Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #21 Jun 09 2019 01:25:38
%S 3,12,108,1332,19908,342252,6583788,139380372,3211960068,79950396492,
%T 2137119431148,61065403377012,1858069709657028,60006976422450732,
%U 2050924514408985708,73988085260209757652,2810535115787602525188
%N Column 2 of triangle A127058.
%C Column 0 of triangle A127058 is A000698, the number of shellings of an n-cube, divided by 2^n n!. Column 1 of triangle A127058 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.
%H Vaclav Kotesovec, <a href="/A127059/b127059.txt">Table of n, a(n) for n = 0..400</a>
%F a(0)=3 and for n>0 a(n)=(1/2)*(c(n+3)-3*c(n+2)-sum(a(k)*(c(n+2-k)-c(n+1-k)),k=0..n-1) with c(n)=(2*n)!/(2^n*n!). - _Groux Roland_, Nov 14 2009
%F G.f.: A(x) = (1 - T(0))/x, T(k) = 1 - x*(k+3)/T(k+1) (continued fraction). - Sergei N. Gladkovskii, Dec 13 2011
%F G.f.: 1/x - Q(0)/x, where Q(k)= 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, May 21 2013
%F a(n) ~ 2^(n + 5/2) * n^(n+3) / exp(n). - _Vaclav Kotesovec_, Jan 02 2019
%t A127058[n_, k_]:= A127058[n, k] = If[k==n, n+1, Sum[A127058[j+k, k]* A127058[n-j, k+1], {j,0,n - k - 1}]]; Table[A127058[n+2, 2], {n, 0, 30}] (* _G. C. Greubel_, Jun 09 2019 *)
%o (PARI) c(n)=(2*n)!/(2^n*n!);
%o a(n)=if(n==0, 3, (c(n+3) - 3*c(n+2) - sum(k=0, n-1, a(k)*(c(n+2-k)-c(n+1-k)) ))/2 );
%o vector(20, n, n--; a(n)) \\ _G. C. Greubel_, Jun 09 2019
%o (Sage)
%o @CachedFunction
%o def A127058(n, k):
%o if (k==n): return n+1
%o else: return sum(A127058(j+k, k)*A127058(n-j, k+1) for j in (0..n-k-1))
%o [A127058(n+2,2) for n in (0..30)] # _G. C. Greubel_, Jun 09 2019
%Y Cf. A127058; other columns: A000698, A115974; A127060.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Jan 04 2007