A127058 -id:A127058 - cp's OEIS frontend

A127059 Column 2 of triangle A127058.

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

Offset: 0

Paul D. Hanna, Jan 04 2007

nonn

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.

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

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

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 *)

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

@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

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

A127060 Row sums of triangle A127058.

Original entry on oeis.org

1, 4, 19, 132, 1253, 14808, 206503, 3298552, 59220265, 1179047100, 25767347387, 613141219356, 15780105110605, 436801028784112, 12941788708753999, 408718346076189360, 13707898517284016849, 486640514520848512692

Offset: 0

Paul D. Hanna, Jan 04 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A127058, A127059.

T[n_, k_]:=T[n, k]=If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];
Table[Sum[T[n, j], {j,0,n}], {n,0,20}] (* G. C. Greubel, Jun 08 2019 *)

getT(n, k, T) = if (!T[n+1,k+1], T[n+1, k+1] = sum(j=0, n-k-1, getT(j+k, k, T)*getT(n-j, k+1, T))); T[n+1, k+1];
tabl(nn) = {my(T = matrix(nn+1, nn+1)); for (i=1, nn+1, T[i, i] = i); for (i=0, nn, for (j=0, i, T[i+1, j+1] = getT(i, j, T); ); ); T; } /* A127059 */
lista(nn) = {my(T = tabl(nn)); vector(nn, k, vecsum(T[k, ]));}
lista(20) \\ Michel Marcus, Jun 09 2019

@CachedFunction
def T(n, k):
    if (k==n): return n+1
    else: return sum(T(j+k, k)*T(n-j, k+1) for j in (0..n-k-1))
def a(n): return sum(T(n,j) for j in (0..n))
[a(n) for n in (0..20)] # G. C. Greubel, Jun 08 2019

a(17) corrected by G. C. Greubel, Jun 08 2019

cp's OEIS Frontend

A127059 Column 2 of triangle A127058.

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