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
Examples
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],...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Programs
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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 *)
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PARI
T(n,k)=if(n
Formula
T(n,k) = Sum_{j=0..k} T(n-k,j)*C(k,j)^2.
Comments