A290533 Numerator of 2*n*(2*n+1) B_{2*n}, where B_n are the Bernoulli numbers.

0, 1, -2, 1, -12, 25, -1382, 245, -28936, 131601, -2444554, 9399643, -4727281820, 1000713051, -332492454406, 43079206380025, -1356840503254192, 1533724275728365, -157891629318320864238, 723708496718865073, -1044330873985796488204

Offset: 0

Author

Seiichi Manyama, Aug 05 2017

Keywords

• sign
• frac

Comments

In 1997, Matiyasevich found the following identity;

(n+2) * Sum_{k=2..n-2} B_k*B_{n-k} - 2 * Sum_{k=2..n-2} binomial(n+2, k)*B_k*B_{n-k} = n*(n+1)*B_n for n > 3.

Examples

			B_n gives the sequence 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, ...
n*(n+1)*B_n gives the sequence 0, -1, 1, 0, -2/3, 0, 1, 0, -12/5, 0, 25/3, 0, -1382/35, 0, 245, 0, -28936/15, ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..314
• Y. Matiyasevich, Identities with Bernoulli numbers, 1997.
• H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, arXiv:math/0407363 [math.NT], 2004.

Crossrefs

Cf. A002427/A006955.