A306228 Numerators of the even moments of the standard V-monotone Gaussian distribution (see the reference in 'Links', Section 5).

1, 1, 4, 28, 278, 3564, 55928, 1037708, 22217720, 539070560, 14616331912, 437960845728, 14370870516352, 512497731949840, 19736969633949568, 816329819676996352, 36089654605723837664, 1698341924904555647808, 84761545323838638225152, 4471847161631552852257472

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Adrian Dacko, V-monotone independence, arXiv:1901.06342 [math.FA], 2019.

Adrian Dacko, Java program.

Formula

a(n) = b(n,n+1), where the term b(n,k) is defined recursively as follows:

For any nonnegative integers n and 1 <= k <= n+2, we have

b(n+1, k) = Sum_{m=0..n} Sum_{s=L1(k,m)..L2(k,m)} binomial(k-1,s)*binomial(n+2-k,m+1-s)*(delta(s,0)*b(m,1) + Sum_{r=1..s} b(m,r))*b(n-m,k-s),

where L1(k,m) = max(0, (m+k)-(n+1)), L2(k,m) = min(k-1,m+1), and delta(s,0) is the Kronecker delta (see the referrence in 'Links', Lemma 5.4).

Prog

(Java) // See links.

Crossrefs

Sequence in context: A128318 A032274 A182964 * A178599 A369088 A007559

Adjacent sequences: A306225 A306226 A306227 * A306229 A306230 A306231

Keyword

frac,nonn

Author

Adrian Dacko, Jan 30 2019

Status

approved