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Central terms of the triangle of central factorial numbers (A008955).
5

%I #11 Feb 19 2022 12:39:03

%S 1,5,273,44473,14739153,8261931405,7026231453265,8439654758970225,

%T 13611213226804376865,28383081191068916580565,

%U 74326386672885754888959569,238812235698229573137588936105,923793013650701305781038113833585,4235104161629281000608041716747118685

%N Central terms of the triangle of central factorial numbers (A008955).

%C a(n) = A008955(2*n,n).

%H Reinhard Zumkeller, <a href="/A234324/b234324.txt">Table of n, a(n) for n = 0..120</a>

%F a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 30.472614208903344955058219429421999299236787591960717... and c = 0.27436634881777520262458169083560234658... - _Vaclav Kotesovec_, Aug 28 2017

%p b:= proc(n, k) option remember; `if`(k=0, 1,

%p add(b(j-1, k-1)*j^2, j=1..n))

%p end:

%p a:= n-> b(2*n, n):

%p seq(a(n), n=0..14); # _Alois P. Heinz_, Feb 19 2022

%t Flatten[{1, Table[Coefficient[Expand[Product[1 + k^2*x, {k, 0, 2*n}]], x^n], {n, 1, 15}]}] (* _Vaclav Kotesovec_, Aug 28 2017 *)

%t Table[Sum[(-1)^(n-j) * StirlingS1[2*n+1, 2*n+1-j] * StirlingS1[2*n+1, j+1], {j, 0, 2*n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Aug 28 2017 *)

%o (Haskell)

%o a234324 n = a008955 (2 * n) n

%Y Cf. A008955, A129505.

%K nonn

%O 0,2

%A _Reinhard Zumkeller_, Dec 24 2013