A127060 -id:A127060 - 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

A127058 Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.

Original entry on oeis.org

1, 2, 2, 10, 6, 3, 74, 42, 12, 4, 706, 414, 108, 20, 5, 8162, 5058, 1332, 220, 30, 6, 110410, 72486, 19908, 3260, 390, 42, 7, 1708394, 1182762, 342252, 57700, 6750, 630, 56, 8, 29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9, 576037442

Offset: 0

Paul D. Hanna, Jan 04 2007

Column 0 is A000698, the number of shellings of an n-cube, divided by 2^n n!.

Column 1 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

			Other recurrences exist, as shown by:
column 0 = A000698: T(n,0) = (2n+1)!! - Sum_{k=1..n} (2k-1)!!*T(n-k,0);
column 1 = A115974: T(n,1) = T(n+1,0) - Sum_{k=0..n-1} T(k,1)*T(n-k,0).
Illustrate the recurrence:
T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) (n > k >= 0)
at column k=1:
T(2,1) = T(1,1)*T(2,2) = 2*3 = 6;
T(3,1) = T(1,1)*T(3,2) + T(2,1)*T(2,2) = 2*12 + 6*3 = 42;
T(4,1) = T(1,1)*T(4,2) + T(2,1)*T(3,2) + T(3,1)*T(2,2) = 2*108 + 6*12 + 42*3 = 414;
at column k=2:
T(3,2) = T(2,2)*T(3,3) = 3*4 = 12;
T(4,2) = T(2,2)*T(4,3) + T(3,2)*T(3,3) = 3*20 + 12*4 = 108;
T(5,2) = T(2,2)*T(5,3) + T(3,2)*T(4,3) + T(4,2)*T(3,3) = 3*220 + 12*20 + 108*4 = 1332.
Triangle begins:
         1;
         2,        2;
        10,        6,       3;
        74,       42,      12,       4;
       706,      414,     108,      20,      5;
      8162,     5058,    1332,     220,     30,     6;
    110410,    72486,   19908,    3260,    390,    42,   7;
   1708394,  1182762,  342252,   57700,   6750,   630,  56,  8;
  29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9; ...

G. C. Greubel, Rows n = 0..15 of triangle, flattened

Columns: A000698, A115974, A127059.
Row sums: A127060.
Cf. A001147 ((2n-1)!!).

T[n_,k_]:= If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

{T(n,k)=if(n==k,n+1,sum(j=0,n-k-1,T(j+k,k)*T(n-j,k+1)))}

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))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 03 2019

cp's OEIS Frontend

A127059 Column 2 of triangle A127058.

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