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.

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

0,2

Comments

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.

Links

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

Example

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; ...

Mathematica

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 *)

Prog

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

Crossrefs

Columns: A000698, A115974, A127059.

Row sums: A127060.

Cf. A001147 ((2n-1)!!).

Sequence in context: A083457 A163808 A223126 * A242002 A094359 A293060

Adjacent sequences: A127055 A127056 A127057 * A127059 A127060 A127061

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 04 2007

Status

approved