A124376 Number triangle with column k generated by x^k*(1+2*k*x+C(k,2)*x^2)/(1-x)^(k+1).

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 19, 10, 1, 1, 13, 37, 37, 13, 1, 1, 16, 61, 92, 61, 16, 1, 1, 19, 91, 185, 185, 91, 19, 1, 1, 22, 127, 326, 440, 326, 127, 22, 1, 1, 25, 169, 525, 896, 896, 525, 169, 25, 1, 1, 28, 217, 792, 1638, 2072, 1638, 792, 217, 28, 1

Offset

0,5

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Formula

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

T(n,k) = T(n,n-k).

Example

Triangle begins
  1,
  1,  1,
  1,  4,  1,
  1,  7,  7,   1,
  1, 10, 19,  10,   1,
  1, 13, 37,  37,  13,  1,
  1, 16, 61,  92,  61, 16,  1,
  1, 19, 91, 185, 185, 91, 19, 1

Mathematica

A124376[n_, k_] := Sum[Binomial[k, k-j]*Binomial[n-j, k]*Binomial[2, j], {j, 0, n}];
Table[A124376[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 21 2025 *)

Prog

(PARI) C(i, j) =binomial(i, j);
T(n, k) = if (k<=n, sum(j=0, n, C(k, k-j)*C(n-j, k)*C(2, j)));
row(n) = vector(n+1, k, T(n, k-1));
for (n=0, 10, print(row(n))) \\ Michel Marcus, Feb 19 2025

Crossrefs

Columns include A016777, A003215, A096000.

Cf. A158920.

Sequence in context: A157172 A131060 A350512 * A047671 A081577 A146986

Adjacent sequences: A124373 A124374 A124375 * A124377 A124378 A124379

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 28 2006

Extensions

More terms from Michel Marcus, Feb 19 2025

Status

approved