cp's OEIS Frontend

A127059 Column 2 of triangle A127058.

%I A127059 #25 Jan 26 2025 17:28:08
%S A127059 3,12,108,1332,19908,342252,6583788,139380372,3211960068,79950396492,
%T A127059 2137119431148,61065403377012,1858069709657028,60006976422450732,
%U A127059 2050924514408985708,73988085260209757652,2810535115787602525188
%N A127059 Column 2 of triangle A127058.
%C A127059 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 A127059 Vaclav Kotesovec, <a href="/A127059/b127059.txt">Table of n, a(n) for n = 0..400</a>
%F A127059 a(0) = 3 and for n>0 a(n) = (1/2)*(c(n+3)-3*c(n+2)-Sum_{k=0..n-1} a(k)*(c(n+2-k)-c(n+1-k))) with c(n) = (2*n)!/(2^n*n!). - _Groux Roland_, Nov 14 2009
%F A127059 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 A127059 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 A127059 a(n) ~ 2^(n + 5/2) * n^(n+3) / exp(n). - _Vaclav Kotesovec_, Jan 02 2019
%t A127059 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 A127059 (PARI) c(n)=(2*n)!/(2^n*n!);
%o A127059 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 A127059 vector(20, n, n--; a(n)) \\ _G. C. Greubel_, Jun 09 2019
%o A127059 (Sage)
%o A127059 @CachedFunction
%o A127059 def A127058(n, k):
%o A127059     if (k==n): return n+1
%o A127059     else: return sum(A127058(j+k, k)*A127058(n-j, k+1) for j in (0..n-k-1))
%o A127059 [A127058(n+2,2) for n in (0..30)] # _G. C. Greubel_, Jun 09 2019
%Y A127059 Cf. A127058; other columns: A000698, A115974; A127060.
%K A127059 nonn
%O A127059 0,1
%A A127059 _Paul D. Hanna_, Jan 04 2007