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Search: a070055 -id:a070055
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A070050
Number of Bottleneck-Monge matrices with 2 rows. In the formula below, P = 2.
+10
8
4, 12, 33, 87, 223, 559, 1375, 3327, 7935, 18687, 43519, 100351, 229375, 520191, 1171455, 2621439, 5832703, 12910591, 28442623, 62390271, 136314879, 296747007, 643825663, 1392508927, 3003121663, 6459228159, 13857980415, 29662117887, 63350767615
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A Bottleneck-Monge matrix is a {0,1} matrix A in which, for every i < j and k < l, max(A[i,l], A[j,k]) <= max(A[i,k], A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).
FORMULA
a(P, N) = 1 + Sum_{n=1..N, p=1..P} (C(N, n) * C(P, p) * K(p, n)),
where:
C(i, j) = binomial(i,j),
K(p, n) = Sum_{i=1..n} (T(p, n, i)),
T(1, n, 1) = 1,
T(1, n, i) = 0, for i > 1,
T(p, n, i) = Sum_{j=i..n, k=1..i} T(p-1, j, k).
a(n) = (n^2 + 3*n + 16)*2^(n - 3) - 1. - Vaclav Kotesovec, Aug 15 2013
From Colin Barker, Sep 10 2017: (Start)
G.f.: x*(4 - 16*x + 21*x^2 - 8*x^3) / ((1 - x)*(1 - 2*x)^3).
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4) for n>4.
(End)
MAPLE
A:=proc(P, N)return 1+add(add(binomial(N, n)*binomial(P, p)*K(p, n), p=1..P), n=1..N):end:
K:=proc(p, n)global k:if(not type(k[p, n], integer))then k[p, n]:=add(T(p, n, i), i=1..n):fi:return k[p, n]:end:
T:=proc(p, n, i)global t:if(not type(t[p, n, i], integer))then if(p=1 and i=1)then t[p, n, i]:=1:elif(p=1)then t[p, n, i]:=0:else t[p, n, i]:=add(add(T(p-1, j, k), k=1..i), j=i..n):fi:fi:return t[p, n, i]:end:
seq(A(2, n), n=1..20); # Nathaniel Johnston, Apr 13 2011
MATHEMATICA
Table[(n^2 + 3*n + 16)*2^(n-3) - 1, {n, 1, 30}] (* Vaclav Kotesovec, Aug 15 2013 *)
PROG
(PARI) Vec(x*(4 - 16*x + 21*x^2 - 8*x^3) / ((1 - x)*(1 - 2*x)^3) + O(x^40)) \\ Colin Barker, Sep 10 2017
CROSSREFS
Cf. A070051 (P=3), A070052 (P=4), A070053 (P=5), A070054 (P=6), A070055 (P=7), A070054 (P=8), A070057 (P=9).
KEYWORD
nonn,easy
AUTHOR
Pascal Prea (pascal.preq(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
Edited by N. J. A. Sloane, Jan 25 2011
a(10)-a(29) and corrected recurrence from Nathaniel Johnston, Apr 13 2011
STATUS
approved
A070051
Number of Bottleneck-Monge matrices with 3 rows. In the formula below, P=3.
+10
8
8, 33, 117, 384, 1194, 3558, 10238, 28606, 77950, 207870, 543998, 1400318, 3552254, 8894462, 22011902, 53903358, 130744318, 314376190, 749928446, 1775894526, 4177264638, 9764863998, 22695378942, 52466548734, 120686903294, 276320747518, 629900574718
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Bottleneck-Monge matrices are {0,1} matrices A in which, for every i<j and k<l, max(A[i,l],A[j,k]) <= max(A[i,k],A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
FORMULA
a(N) = a(3, N), where a(P, N) is defined recursively in A070050.
Empirical G.f.: -x*(32*x^5-140*x^4+213*x^3-154*x^2+55*x-8) / ((x-1)*(2*x-1)^5). - Colin Barker, Feb 28 2013
CROSSREFS
Cf. A070050, A070052, A070053, A070054, A070055, A070056, A070057.
KEYWORD
nonn,easy
AUTHOR
Pascal Prea (pascal.prea(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
a(10) - a(27) from Nathaniel Johnston, Apr 13 2011
STATUS
approved
A070052
Number of Bottleneck-Monge matrices with 4 rows.
+10
8
16, 87, 384, 1516, 5535, 19030, 62347, 196301, 597725, 1768733, 5105853, 14423037, 39968765, 108884733, 292116989, 772923389, 2019573757, 5216813053, 13334904829, 33758183421, 84702265341, 210777210877, 520498839549, 1276178857981, 3108165910525, 7522859614205, 18101546516477
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Bottleneck-Monge matrices are {0,1} matrices A in which, for every i<j and k<l, max(A[i,l],A[j,k]) <= max(A[i,k],A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
FORMULA
a(N) = a(4, N), where a(P, N) is defined recursively in A070050.
Empirical G.f.: -x*(128*x^7-799*x^6+1835*x^5-2199*x^4+1542*x^3-647*x^2+153*x-16) / ((x-1)*(2*x-1)^7). - Colin Barker, Feb 28 2013
CROSSREFS
Cf. A070050, A070051, A070053, A070054, A070055, A070056, A070057.
KEYWORD
nonn,easy
AUTHOR
Pascal Prea (pascal.prea(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
a(10) - a(27) from Nathaniel Johnston, Apr 13 2011
STATUS
approved
A070054
Number of Bottleneck-Monge matrices with 6 rows.
+10
8
64, 559, 3558, 19030, 90398, 393133, 1595475, 6121871, 22418665, 78909805, 268421215, 886226507, 2849922043, 8952215067, 27534649147, 83092274171, 246448262139, 719487237115, 2070217930747, 5877572837371, 16481768407035, 45689773670395, 125310468063227, 340264608595963, 915344114450427, 2440847110832123
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Bottleneck-Monge matrices are {0,1} matrices A in which, for every i<j and k<l, max(A[i,l],A[j,k]) <= max(A[i,k],A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
FORMULA
a(N) = a(6, N), where a(P, N) is defined recursively in A070050.
Empirical G.f.: -x*(2048*x^11-21010*x^10+81464*x^9-175106*x^8+241444*x^7-229084*x^6+154777*x^5-75284*x^4+26086*x^3-6189*x^2+913*x-64) / ((x-1)*(2*x-1)^11). - Colin Barker, Feb 28 2013
CROSSREFS
Cf. A070050, A070051, A070052, A070054, A070055, A070056, A070057.
KEYWORD
nonn
AUTHOR
Pascal Prea (pascal.prea(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
a(10) - a(26) from Nathaniel Johnston, Apr 13 2011
STATUS
approved
A070057
Number of Bottleneck-Monge matrices with 9 rows.
+10
8
512, 7935, 77950, 597725, 3887592, 22418665, 117789230, 574209267, 2630933289, 11438763414, 47540271657, 189960445332, 733160014779, 2743585958976, 9986068392201, 35447227965486, 122988982790340, 417920043272208, 1393150860920936, 4562718944632792, 14700625316815800, 46648299847265144, 145938710212960504
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Bottleneck-Monge matrices are {0,1} matrices A in which, for every i<j and k<l, max(A[i,l],A[j,k]) <= max(A[i,k],A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
FORMULA
a(N) = a(9, N), where a(P, N) is defined recursively in A070050.
G.f.: x*(512 - 9985*x + 96161*x^2 - 607903*x^3 + 2821517*x^4 - 10164757*x^5 + 29260931*x^6 - 68263237*x^7 + 129682732*x^8 - 200278047*x^9 + 249653465*x^10 - 247990203*x^11 + 192418577*x^12 - 113113161*x^13 + 48011519*x^14 - 13562349*x^15 + 2175308*x^16 - 131072*x^17) / ((1 - x)*(1 - 2*x)^17) (conjectured). - Colin Barker, Sep 10 2017
CROSSREFS
Cf. A070050, A070051, A070052, A070053, A070054, A070055, A070056.
KEYWORD
nonn
AUTHOR
Pascal Prea (pascal.prea(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
a(9)-a(23) from Nathaniel Johnston, Apr 13 2011
STATUS
approved
A070053
Number of Bottleneck-Monge matrices with 5 rows.
+10
7
32, 223, 1194, 5535, 23225, 90398, 331561, 1158554, 3887592, 12603092, 39659340, 121593628, 364327996, 1069496444, 3082349564, 8737174524, 24395286524, 67182399484, 182689947644, 491040063484, 1305703776252, 3437454229500, 8965943918588, 23184233594876, 59466277191676, 151373599997948, 382587234156540
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Bottleneck-Monge matrices are {0,1} matrices A in which, for every i<j and k<l, max(A[i,l],A[j,k]) <= max(A[i,k],A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
FORMULA
a(N) = a(5, N), where a(P, N) is defined recursively in A070050.
Empirical G.f.: -x*(512*x^9-4204*x^8+12965*x^7-21713*x^6+22623*x^5-15536*x^4+7137*x^3-2141*x^2+385*x-32) / ((x-1)*(2*x-1)^9). - Colin Barker, Feb 28 2013
CROSSREFS
Cf. A070050, A070051, A070052, A070054, A070055, A070056, A070057.
KEYWORD
nonn
AUTHOR
Pascal Prea (pascal.prea(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
a(10) - a(27) from Nathaniel Johnston, Apr 13 2011
STATUS
approved
A070056
Number of Bottleneck-Monge matrices with 8 rows.
+10
7
256, 3327, 28606, 196301, 1158554, 6121871, 29688844, 134361100, 574209267, 2337788346, 9128782573, 34372098576, 125327730017, 444081743234, 1533640995983, 5174881667481, 17096446731321, 55402030270777, 176377009761145, 552397160169465, 1704039417961465, 5183132984363513, 15559823619750905, 46141145945628665
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Bottleneck-Monge matrices are {0,1} matrices A in which, for every i<j and k<l, max(A[i,l],A[j,k]) <= max(A[i,k],A[j,l]).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..400
FORMULA
a(N) = a(8, N), where a(P, N) is defined recursively in A070050.
G.f.: x*(256 - 4609*x + 40669*x^2 - 232695*x^3 + 961183*x^4 - 3017869*x^5 + 7388387*x^6 - 14256058*x^7 + 21694227*x^8 - 25838259*x^9 + 23693517*x^10 - 16267523*x^11 + 7986763*x^12 - 2589705*x^13 + 474491*x^14 - 32768*x^15) / ((1 - x)*(1 - 2*x)^15) (conjectured). - Colin Barker, Sep 10 2017
CROSSREFS
Cf. A070050, A070051, A070052, A070053, A070054, A070055, A070057.
KEYWORD
nonn
AUTHOR
Pascal Prea (pascal.prea(AT)lim.univ-mrs.fr), Apr 18 2002
EXTENSIONS
a(10)-a(24) from Nathaniel Johnston, Apr 13 2011
STATUS
approved
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