A265603 -id:A265603

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A265314

Triangle read by rows, the numerators of the Bell transform of B(n,1) where B(n,x) are the Bernoulli polynomials.

+10
3

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 0, 17, 3, 1, 0, -1, 5, 65, 5, 1, 0, 0, 7, 55, 175, 15, 1, 0, 1, -7, 2023, 245, 385, 21, 1, 0, 0, -38, 49, 34181, 595, 371, 14, 1, 0, -1, 3, -14351, 973, 56567, 525, 217, 18, 1, 0, 0, 99, -19, 10637, 13601, 208859, 2415, 355, 45, 1

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

OFFSET

0,9

COMMENTS

For the definition of the Bell transform see A264428 and the link given there.

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

0, 1,

0, 1, 1,

0, 1, 3, 1,

0, 0, 17, 3, 1,

0, -1, 5, 65, 5, 1,

0, 0, 7, 55, 175, 15, 1,

0, 1, -7, 2023, 245, 385, 21, 1,

0, 0, -38, 49, 34181, 595, 371, 14, 1,

0, -1, 3, -14351, 973, 56567, 525, 217, 18, 1.

MAPLE

A265314_triangle := proc(n) local B, C, k;

B := BellMatrix(x -> bernoulli(x, 1), n); # see A264428

for k from 1 to n do

C := LinearAlgebra:-Row(B, k):

print(seq(numer(C[j]), j=1..k))

od end:

A265314_triangle(10);

MATHEMATICA

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

rows = 12;

B = BellMatrix[Function[x, BernoulliB[x, 1]], rows];

Table[B[[n, k]] // Numerator, {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, from Maple *)

CROSSREFS

Cf. A265315 for the denominators, A265602 and A265603 for B(2n,1).

Cf. A027641 and A164555 (column 1).

KEYWORD

sign,tabl,frac

AUTHOR

Peter Luschny, Jan 22 2016

STATUS

approved

A265315

Triangle read by rows, the denominators of the Bell transform of B(n,1) where B(n,x) are the Bernoulli polynomials.

+10
3

1, 1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 1, 12, 1, 1, 1, 30, 6, 12, 1, 1, 1, 1, 90, 8, 12, 2, 1, 1, 42, 20, 360, 8, 12, 2, 1, 1, 1, 315, 45, 720, 6, 6, 1, 1, 1, 30, 7, 3780, 20, 240, 2, 2, 1, 1, 1, 1, 350, 7, 756, 32, 240, 4, 2, 2, 1, 1, 66, 12, 6300, 1512, 6048, 96, 240, 4, 1, 2, 1

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

OFFSET

0,5

COMMENTS

For the definition of the Bell transform see A264428 and the link given there.

LINKS

Table of n, a(n) for n=0..77.

EXAMPLE

1, 1,

1, 2, 1,

1, 6, 2, 1,

1, 1, 12, 1, 1,

1, 30, 6, 12, 1, 1,

1, 1, 90, 8, 12, 2, 1,

1, 42, 20, 360, 8, 12, 2, 1,

1, 1, 315, 45, 720, 6, 6, 1, 1,

1, 30, 7, 3780, 20, 240, 2, 2, 1, 1,

1, 1, 350, 7, 756, 32, 240, 4, 2, 2, 1.

MAPLE

A265315_triangle := proc(n) local B, C, k;

B := BellMatrix(x -> bernoulli(x, 1), n); # see A264428

for k from 1 to n do

C := LinearAlgebra:-Row(B, k):

print(seq(denom(C[j]), j=1..k))

od end:

A265315_triangle(12);

MATHEMATICA

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

rows = 12;

B = BellMatrix[Function[x, BernoulliB[x, 1]], rows];

Table[B[[n, k]] // Denominator, {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, from Maple *)

CROSSREFS

Cf. A265314 for the numerators, A265602 and A265603 for B(2n,1).

Cf. A027642 (column 1).

KEYWORD

nonn,tabl,frac

AUTHOR

Peter Luschny, Jan 22 2016

STATUS

approved

A265602

Triangle read by rows, the numerators of the Bell transform of B(2n,1) where B(n,x) are the Bernoulli polynomials.

+10
3

1, 0, 1, 0, 1, 1, 0, -1, 1, 1, 0, 1, -1, 1, 1, 0, -1, 4, 1, 5, 1, 0, 5, -163, 47, 7, 5, 1, 0, -691, 191, -109, 11, 7, 7, 1, 0, 7, -1431809, 6869, -253, 1, 119, 14, 1, 0, -3617, 130168, -7728013, 2659, -83, 11, 77, 6, 1

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

OFFSET

0,18

COMMENTS

For the definition of the Bell transform see A264428 and the link given there.

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

0, 1,

0, 1, 1,

0, -1, 1, 1,

0, 1, -1, 1, 1,

0, -1, 4, 1, 5, 1,

0, 5, -163, 47, 7, 5, 1,

0, -691, 191, -109, 11, 7, 7, 1,

0, 7, -1431809, 6869, -253, 1, 119, 14, 1,

0, -3617, 130168, -7728013, 2659, -83, 11, 77, 6, 1.

MAPLE

A265602_triangle := proc(n) local B, C, k;

B := BellMatrix(x -> bernoulli(2*x, 1), n); # see A264428

for k from 1 to n do

C := LinearAlgebra:-Row(B, k):

print(seq(numer(C[j]), j=1..k))

od end:

A265602_triangle(10);

MATHEMATICA

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

rows = 12;

B = BellMatrix[BernoulliB[2#, 1]&, rows];

Table[B[[n, k]] // Numerator, {n, 1, rows}, {k, 1, n}] // Flatten (*~, from Maple *) ~~~

CROSSREFS

Cf. A265603 for the denominators, A265314 and A265315 for B(n,1).

Cf. A000367 (column 1).

KEYWORD

sign,tabl,frac

AUTHOR

Peter Luschny, Jan 21 2016

STATUS

approved

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