A369134 -id:A369134

Search: a369134 -id:a369134

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page 1

A369121

a(n) = A369134(n, n).

+20
2

-1, 1, -1, 3, -1, 5, -15, 105, -21, 105, -315, 2475, -99, 1001, -1001, 15015, -105105, 23205, -663, 1322685, -101745, 5595975, -2611455, 60063465, -1225785, 15935205, -51555075, 4686825, -5368545, 40970475, -40970475, 302703526125

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..31.

CROSSREFS

Cf. A369134, A369122.

KEYWORD

sign

AUTHOR

Peter Luschny, Jan 14 2024

STATUS

approved

A369122

a(n) = A369134(n, 2).

+20
2

0, 0, -1, 7, -14, 693, -30030, 4150146, -21441420, 3508377873, -425496301230, 163608770318994, -381271515244620, 263655717130655850, -20916990073196036820, 28579221265797444862620, -20735968190374168899967800, 535753174085515196009964765, -2008476391973175078407803350

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..18.

CROSSREFS

Cf. A369134, A369121.

KEYWORD

sign

AUTHOR

Peter Luschny, Jan 14 2024

STATUS

approved

A369120

a(n) = Sum_{k=0..n} A369134(n, k).

+20
1

-1, 1, -1, 10, -21, 1050, -45606, 6306300, -32585553, 5332033850, -646674981498, 248655336029100, -579463114572870, 400708622091878100, -31790012531476579380, 43435207772044760997000, -31514892593265599765292045, 814247185935070977732893250

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

See A369134 for comments.

LINKS

Table of n, a(n) for n=0..17.

FORMULA

a(n) / A369135(n) = Bernoulli(2*n).

MATHEMATICA

A368846[n_, k_] := If[k == 0, Boole[n == 0], (-1)^(n + k) 2 Binomial[2 k - 1, n] Binomial[2 n + 1, 2 k]];

Map[Total[# LCM @@ Denominator[#]]&, MapIndexed[(-1)^First[#2] Take[#, First[#2]]&, Inverse[PadRight[Table[A368846[n, k], {n, 0, 20}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 16 2024 *)

CROSSREFS

Cf. A369134, A000367/A002445 (Bernoulli(2n)).

KEYWORD

sign

AUTHOR

Peter Luschny, Jan 14 2024

STATUS

approved

A368846

Triangle read by rows: T(n, k) = (-1)^(n + k)*2*binomial(2*k - 1, n)* binomial(2*n + 1, 2*k) for k > 0, and k^n for k = 0.

+10
6

1, 0, 6, 0, 0, 30, 0, 0, -70, 140, 0, 0, 0, -840, 630, 0, 0, 0, 924, -6930, 2772, 0, 0, 0, 0, 18018, -48048, 12012, 0, 0, 0, 0, -12870, 216216, -300300, 51480, 0, 0, 0, 0, 0, -350064, 2042040, -1750320, 218790, 0, 0, 0, 0, 0, 184756, -5542680, 16628040, -9699690, 923780

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The row sums of the inverse triangle (A368847/A368848) are the unsigned Bernoulli numbers |B(2n)|. To get the signed Bernoulli numbers B(2n), one only needs to change the sign factor in the definition from (-1)^(n + k) to (-1)^(n + 1).

Conjecture: |Sum_{j=0..k} T(k + j, k)| = A229580(k + 1) for k >= 0.

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.

EXAMPLE

[0] [1]

[1] [0, 6]

[2] [0, 0, 30]

[3] [0, 0, -70, 140]

[4] [0, 0, 0, -840, 630]

[5] [0, 0, 0, 924, -6930, 2772]

[6] [0, 0, 0, 0, 18018, -48048, 12012]

[7] [0, 0, 0, 0, -12870, 216216, -300300, 51480]

[8] [0, 0, 0, 0, 0, -350064, 2042040, -1750320, 218790]

MATHEMATICA

A368846[n_, k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];

Table[A368846[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 08 2024 *)

PROG

(SageMath)

def A368846(n, k):

if k == 0: return k^n

if k > n: return 0

return (-1)^(n + k)*2*binomial(2*k - 1, n)*binomial(2*n + 1, 2*k)

for n in range(10): print([A368846(n, k) for k in range(n+1)])

CROSSREFS

Cf. A368847/A368848 (inverse), A369134, A369135, A002457 (main diagonal), A000367/A002445 (Bernoulli(2n)), A229580.

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jan 07 2024

STATUS

approved

A369135

a(n) is the lcm of the denominators of the terms in the n-th row of M where M is the inverse of the matrix generated by the triangle A368846.

+10
6

1, 6, 30, 420, 630, 13860, 180180, 5405400, 4594590, 96996900, 1222160940, 40156716600, 6692786100, 281097016200, 1164544781400, 72201776446800, 2084826294901350, 1895296631728500, 222622144044300, 1823275359722817000, 575032998066426900, 129519337183533297000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

See A369134 for comments and formulas.

LINKS

Table of n, a(n) for n=0..21.

MATHEMATICA

A368846[n_, k_] := If[k == 0, Boole[n == 0], (-1)^(n + k) 2 Binomial[2 k - 1, n] Binomial[2 n + 1, 2 k]];

LCM @@@ Denominator[MapIndexed[Take[#, First[#2]]&, Inverse[PadRight[Table[ A368846[n, k], {n, 0, 25}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 14 2024 *)

PROG

(SageMath)

M = matrix(ZZ, 32, 32, A368846).inverse()

def A369135(n): return lcm(M[n][k].denominator() for k in range(n + 1))

print([A369135(n) for n in range(21)])

CROSSREFS

Cf. A369134, A368846.

KEYWORD

sign

AUTHOR

Peter Luschny, Jan 14 2024

STATUS

approved

A368848

Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846.

+10
4

1, 1, 6, 1, 1, 30, 1, 1, 60, 140, 1, 1, 45, 105, 630, 1, 1, 20, 140, 252, 2772, 1, 1, 6, 14, 1260, 693, 12012, 1, 1, 900, 2100, 945, 5940, 10296, 51480, 1, 1, 3, 1, 945, 189, 1287, 6435, 218790, 1, 1, 100, 700, 420, 660, 12012, 780, 145860, 923780

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The row sums of the triangle, seen in its rational form A368847(n)/ A368848(n), are the unsigned Bernoulli numbers |B(2n)|. To get the signed Bernoulli numbers B(2n), one only needs to change the sign factor in the definition of A368846 from (-1)^(n + k) to (-1)^(n + 1).

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.

EXAMPLE

Triangle starts:

[0] [1]

[1] [1, 6]

[2] [1, 1, 30]

[3] [1, 1, 60, 140]

[4] [1, 1, 45, 105, 630]

[5] [1, 1, 20, 140, 252, 2772]

[6] [1, 1, 6, 14, 1260, 693, 12012]

[7] [1, 1, 900, 2100, 945, 5940, 10296, 51480]

[8] [1, 1, 3, 1, 945, 189, 1287, 6435, 218790]

MATHEMATICA

A368846[n_, k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];

Denominator[MapIndexed[Take[#, First[#2]]&, Inverse[PadRight[Table[ A368846[n, k], {n, 0, 10}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 08 2024 *)

PROG

(SageMath)

M = matrix(ZZ, 10, 10, lambda n, k: A368846(n, k) if k <= n else 0)

I = M.inverse()

for n in range(9): print([I[n][k].denominator() for k in range(n+1)])

CROSSREFS

Cf. A368846 (inverse), A368847 (numerator), A002457 (main diagonal), A369134, A369135, A000367/A002445 (Bernoulli(2n)).

KEYWORD

nonn,tabl,frac

AUTHOR

Peter Luschny, Jan 07 2024

STATUS

approved

A368847

Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.

+10
3

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 1, 1, 17, 1, 1, 0, 0, 691, 691, 59, 41, 5, 1, 0, 0, 14, 2, 359, 8, 4, 1, 1, 0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1, 0, 0, 43867, 43867, 750167, 6583, 943, 1129, 217, 2, 1, 0, 0, 1222277, 174611, 627073, 1540967, 28399, 53, 47, 23, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,19

COMMENTS

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.

Peter Luschny, Illustrating the polynomials.

EXAMPLE

[0] [1]

[1] [0, 1]

[2] [0, 0, 1]

[3] [0, 0, 1, 1]

[4] [0, 0, 1, 1, 1]

[5] [0, 0, 1, 3, 1, 1]

[6] [0, 0, 1, 1, 17, 1, 1]

[7] [0, 0, 691, 691, 59, 41, 5, 1]

[8] [0, 0, 14, 2, 359, 8, 4, 1, 1]

[9] [0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1]

MATHEMATICA

A368846[n_, k_]:=If[k==0, Boole[n==0], (-1)^(n+k)2Binomial[2k-1, n]Binomial[2n+1, 2k]];

Numerator[MapIndexed[Take[#, First[#2]]&, Inverse[PadRight[Table[A368846[n, k], {n, 0, 10}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 08 2024 *)

PROG

(SageMath)

M = matrix(ZZ, 10, 10, lambda n, k: A368846(n, k) if k <= n else 0)

I = M.inverse()

for n in range(9): print([I[n][k].numerator() for k in range(n+1)])

CROSSREFS

Cf. A368846 (inverse), A368848 (denominator), A369134, A369135, A000367/A002445 (Bernoulli(2n)).

KEYWORD

nonn,frac,tabl

AUTHOR

Peter Luschny, Jan 07 2024

STATUS

approved

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