A097084 Triangle, read by rows, where the n-th diagonal equals the n-th row transformed by triangle A008459 (squared binomial coefficients).

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 10, 10, 1, 1, 5, 18, 28, 17, 1, 1, 6, 27, 74, 69, 26, 1, 1, 7, 39, 137, 245, 151, 37, 1, 1, 8, 52, 236, 586, 676, 298, 50, 1, 1, 9, 68, 372, 1194, 2126, 1634, 540, 65, 1, 1, 10, 85, 552, 2322, 5152, 6620, 3578, 913, 82, 1, 1, 11, 105, 777, 3954, 12002, 19292, 18082, 7249, 1459, 101, 1

Offset

0,5

Comments

Row sums form A097085.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,k) = Sum_{j=0..k} T(n-k,j)*C(k,j)^2.

Example

T(8,3) = 236 = (1)*1^2 + (5)*3^2 + (18)*3^2 + (28)*1^2
= Sum_{j=0..3} T(5,j)*C(3,j)^2.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,5,1],
[1,4,10,10,1],
[1,5,18,28,17,1],
[1,6,27,74,69,26,1],
[1,7,39,137,245,151,37,1],
[1,8,52,236,586,676,298,50,1],...

Maple

T:= proc(n, k) option remember;
      `if`(n=k or k=0, 1, `if`(k<0 or k>n, 0,
       add(T(n-k, j)*binomial(k, j)^2, j=0..k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 30 2015

Mathematica

T[_, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; 0 < k < n := T[n, k] = Sum[T[n - k, j]*Binomial[k, j]^2, {j, 0, k}]; T[_, _] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 24 2016 *)

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k || k==0, 1, sum(j=0, n-k, T(n-k, j)*binomial(k, j)^2)))

Crossrefs

Cf. A097085, A008459.

Sequence in context: A258306 A049513 A121207 * A143327 A094954 A083064

Adjacent sequences: A097081 A097082 A097083 * A097085 A097086 A097087

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 23 2004

Status

approved