A103916 Column k=2 sequence (without zero entries) of table A060524.

1, 14, 439, 24940, 2250621, 296266266, 53624576979, 12780684581400, 3880806293223225, 1462807581365269350, 670261417348408188975, 366936357918296751120900, 236559234981486279096163125

Offset

0,2

Comments

a(n) = sum over all multinomials M2(2*(n+1),k), k from {1..p(2*(n+1))} restricted to partitions with exactly two odd and any nonnegative number even parts. p(2*(n+1)) = A000041(2*(n+1)) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*(n+1),k). - Wolfdieter Lang, Aug 07 2007

Links

Table of n, a(n) for n=0..12.

Formula

E.g.f. (with alternating zeros): A(x) = (d^2/dx^2)a(x) with a(x):=(1/(sqrt(1-x^2))*(log(sqrt((1+x)/(1-x))))^2)/2!.

a(n) ~ log(2*n)^2 * 2^(2*n) * n^(2*n + 2) / (exp(2*n)) * (1 + (2*gamma + 6*log(2))/log(2*n) + (gamma^2 + 6*gamma*log(2) + 9*log(2)^2 - Pi^2/2) / log(2*n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 21 2019

Example

Multinomial representation for a(2): partitions of 2*3=6 with two odd parts: (1,5) with A-St position k=2; (3^2) with k=4; (1^2,4) with k=5; (1,2,3) with k=6 and (1^2,2^2) with k=9. The M2 numbers for these partitions are 144, 40, 90, 120, 45, adding up to 439 = a(2).

Mathematica

nmax = 20; Table[(CoefficientList[Series[(4 + 8*x*Log[(1 + x)/(1 - x)] + (1/2 + x^2)*Log[(1 + x)/(1 - x)]^2)/(4*(1 - x^2)^(5/2)), {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Jul 21 2019 *)

Crossrefs

Sequence in context: A269504 A033815 A187358 * A201546 A305115 A282245

Adjacent sequences: A103913 A103914 A103915 * A103917 A103918 A103919

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 24 2005

Status

approved