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A055815 a(n) = T(2*n+3,n), array T as in A055807.

%I A055815 #17 Jul 31 2025 11:55:05
%S A055815 1,15,80,432,2352,12896,71136,394400,2196128,12273648,68811184,
%T A055815 386838480,2179890000,12309739968,69641542848,394643939904,
%U A055815 2239678552640,12727572969680,72415319422992,412470467298032
%N A055815 a(n) = T(2*n+3,n), array T as in A055807.
%H A055815 G. C. Greubel, <a href="/A055815/b055815.txt">Table of n, a(n) for n = 0..250</a>
%F A055815 a(n) = (n+3)*hypergeom([-n-2, n], [2], -1) = Sum_{s=1..n+3} binomial(n+3,s) * binomial(s+n-2,n-1) for n >= 1. - _Petros Hadjicostas_, Feb 13 2021
%p A055815 T:= proc(i, j) option remember;
%p A055815       if j=0 then 1
%p A055815     elif i=0 then 0
%p A055815     else add(add(T(h,m), m=0..j), h=0..i-1)
%p A055815   fi; end:
%p A055815 seq(T(n+3, n), n=0..20); # _G. C. Greubel_, Jan 23 2020
%t A055815 T[i_, j_]:= T[i, j]= If[j==0, 1, If[i==0, 0, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]]; Table[T[n+3, n], {n,0,20}] (* _G. C. Greubel_, Jan 23 2020 *)
%o A055815 (Sage)
%o A055815 @CachedFunction
%o A055815 def T(i, j):
%o A055815     if (j==0): return 1
%o A055815     elif (i==0): return 0
%o A055815     else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
%o A055815 [T(n+3, n) for n in (0..20)] # _G. C. Greubel_, Jan 23 2020
%Y A055815 Apart from the offset the same as A050149. - _R. J. Mathar_, Oct 13 2008
%Y A055815 Cf. A055807, A055809, A055810, A055811, A055816, A055817.
%K A055815 nonn
%O A055815 0,2
%A A055815 _Clark Kimberling_, May 28 2000