A127059 - OEIS

A127059

Column 2 of triangle A127058.

3, 12, 108, 1332, 19908, 342252, 6583788, 139380372, 3211960068, 79950396492, 2137119431148, 61065403377012, 1858069709657028, 60006976422450732, 2050924514408985708, 73988085260209757652, 2810535115787602525188

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

FORMULA

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

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

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

a(n) ~ 2^(n + 5/2) * n^(n+3) / exp(n). - Vaclav Kotesovec, Jan 02 2019

MATHEMATICA

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 *)

PROG

(PARI) c(n)=(2*n)!/(2^n*n!);

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 );

vector(20, n, n--; a(n)) \\ G. C. Greubel, Jun 09 2019

(Sage)

@CachedFunction

def A127058(n, k):

if (k==n): return n+1

else: return sum(A127058(j+k, k)*A127058(n-j, k+1) for j in (0..n-k-1))

[A127058(n+2, 2) for n in (0..30)] # G. C. Greubel, Jun 09 2019

CROSSREFS

Cf. A127058; other columns: A000698, A115974; A127060.