A292901 Triangle read by rows, a generalization of the Bernoulli numbers, the denominators for n>=0 and 0<=k<=n.

1, 1, 2, 1, 2, 6, 1, 2, 3, 1, 1, 2, 12, 3, 30, 1, 2, 24, 9, 20, 1, 1, 2, 48, 54, 80, 10, 42, 1, 2, 96, 324, 8640, 200, 105, 1, 1, 2, 192, 1944, 3840, 36000, 525, 35, 30, 1, 2, 384, 11664, 1244160, 720000, 756000, 3675, 168, 1

Offset

0,3

Comments

See comments in A292900.

Links

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

S. Fukuhara, N. Kawazumi and Y. Kuno, Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface, arXiv:1505.04840 [math.NT], 2015.

L. Kronecker, Ueber die Bernoullischen Zahlen, J. Reine Angew. Math. 94 (1883), 268-269.

Example

Triangle starts:
[0], 1
[1], 1, 2
[2], 1, 2,   6
[3], 1, 2,   3,     1
[4], 1, 2,  12,     3,      30
[5], 1, 2,  24,     9,      20,      1
[6], 1, 2,  48,    54,      80,     10,     42
[7], 1, 2,  96,   324,    8640,    200,    105,    1
[8], 1, 2, 192,  1944,    3840,  36000,    525,   35,  30
[9], 1, 2, 384, 11664, 1244160, 720000, 756000, 3675, 168, 1

Maple

# Function B(n, k) in A292900.
for n from 0 to 9 do seq(denom(B(n, k)), k=0..n) od;

Mathematica

B[0, 0] = 1; B[n_, k_] := Sum[(-1)^(j-n)/(j+1) Binomial[k+1, j+1] Sum[i^n (j-i+1)^(k-n), {i, 0, j}], {j, 0, k}]; Table[B[n, k] // Denominator, {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 14 2019, from Maple *)

Crossrefs

Cf. A292900 (numerators), T(n, n) = A027642(n).

Sequence in context: A049404 A159885 A178803 * A083773 A129116 A096179

Adjacent sequences: A292898 A292899 A292900 * A292902 A292903 A292904

Keyword

nonn,tabl,frac

Author

Peter Luschny, Oct 01 2017

Status

approved