A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)*(j+1)*[(k+1)*(k+2)/2 - j*(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.

1, 1, 1, 1, 7, 7, 1, 31, 139, 139, 1, 127, 1567, 5711, 5711, 1, 511, 15379, 126579, 408354, 408354, 1, 2047, 143527, 2357431, 15333661, 45605881, 45605881, 1, 8191, 1312219, 40769819, 473433344, 2634441290, 7390305396, 7390305396, 1, 32767

Offset

0,5

Comments

Main diagonal is A082162 (with offset). This sequence is derived from column 0 of A102098.

Links

Table of n, a(n) for n=0..37.

Example

T(4,2) = 1567 = 1*6 + 31*10 + 139*9
= T(3,0)*R(0,2) + T(3,1)*R(1,2) + T(3,2)*R(2,2).
Rows begin:
[1],
[1,1],
[1,7,7],
[1,31,139,139],
[1,127,1567,5711,5711],
[1,511,15379,126579,408354,408354],
[1,2047,143527,2357431,15333661,45605881,45605881],...
where the transpose of the recurrence coefficients given by
[R^t](n,k) = (k+1)*((n+1)*(n+2)/2 - k*(k+1)/2) form triangle:
[1],
[3,4],
[6,10,9],
[10,18,21,16],
[15,28,36,36,25],...
which equals the matrix square of the triangle:
[1],
[1,2],
[1,2,3],
[1,2,3,4],
[1,2,3,4,5],...

Mathematica

T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[k == 0, 1, If[n == k, T[n, n-1], Sum[T[n-1, j]*(j+1)*((k+1)*(k+2)/2-j*(j+1)/2), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 15 2014, after PARI *)

Prog

(PARI) {T(n, k)=if(n<k|k<0, 0, if(k==0, 1, if(n==k, T(n, n-1), sum(j=0, k, T(n-1, j)*(j+1)*((k+1)*(k+2)/2-j*(j+1)/2)))))}

Crossrefs

Cf. A082162, A102098, A102317.

Sequence in context: A232650 A365672 A108390 * A144860 A113810 A002161

Adjacent sequences: A102397 A102398 A102399 * A102401 A102402 A102403

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 06 2005

Status

approved