The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A292900

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A292900

Triangle read by rows, a generalization of the Bernoulli numbers, the numerators for n>=0 and 0<=k<=n.
(history; published version)

#20 by Bruno Berselli at Thu Jul 19 08:23:29 EDT 2018

STATUS

proposed

approved

#19 by Jean-François Alcover at Thu Jul 19 08:17:18 EDT 2018

STATUS

editing

proposed

#18 by Jean-François Alcover at Thu Jul 19 07:44:55 EDT 2018

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] // Numerator, {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, from Maple *)

STATUS

approved

editing

#17 by Peter Luschny at Mon Oct 02 03:29:05 EDT 2017

STATUS

editing

approved

#16 by Peter Luschny at Mon Oct 02 03:28:55 EDT 2017

MAPLE

B := (n, k) -> `if`(n = 0, n^k, 1, add(((-1)^(j-n)/(j+1))*binomial(k+1, j+1)*add(i^n*(j-i+1)^(k-n), i=0..j), j=0..k)):

CROSSREFS

Cf. A292901 (denominators), B(n, n) = A164555(n)/A027642(n), A215083.

STATUS

approved

editing

#15 by Peter Luschny at Sun Oct 01 16:20:28 EDT 2017

STATUS

editing

approved

#14 by Peter Luschny at Sun Oct 01 16:20:25 EDT 2017

COMMENTS

The diagonal B(n, n) give gives the Bernoulli numbers B_n = B_n(1). The formula is due to L. Kronecker and the generalization to Fukuhara, Kawazumi and Kuno.

STATUS

approved

editing

#13 by Peter Luschny at Sun Oct 01 16:18:48 EDT 2017

STATUS

proposed

approved

#12 by Peter Luschny at Sun Oct 01 13:22:08 EDT 2017

STATUS

editing

proposed

#11 by Peter Luschny at Sun Oct 01 13:16:46 EDT 2017

KEYWORD

sign,tabl,frac,changed

STATUS

proposed

editing

Discussion

Sun Oct 01

13:17

Peter Luschny: Of course. Thanks Michel!