A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

1, 2, 1, 5, 4, 1, 13, 15, 6, 1, 35, 52, 30, 8, 1, 96, 175, 130, 50, 10, 1, 267, 576, 525, 260, 75, 12, 1, 750, 1869, 2016, 1225, 455, 105, 14, 1, 2123, 6000, 7476, 5376, 2450, 728, 140, 16, 1, 6046, 19107, 27000, 22428, 12096, 4410, 1092, 180, 18, 1

Offset

0,2

Links

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

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively.

E.g.f. of column k: exp(x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 16 2023

Example

Triangle begins:
   1;
   2,   1;
   5,   4,  1;
  13,  15,  6, 1;
  35,  52, 30, 8, 1;
  ...

Maple

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1/2,
      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))
      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n+1$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015
# second program:
A171651:= (n, k)-> binomial(n, k)*add((-1)^(n-k-j)*binomial(n-k, j)*binomial(2*j+1, j+1), j=0..n-k): seq(print(seq(A171651(n, k), k=0..n)), n=0..9);  # Mélika Tebni, Dec 16 2023

Mathematica

b[u_, d_, t_] := b[u, d, t] = If[u == 0 && d == 0, 1/2, Expand[If[u == 0, 0, b[u-1, d, 2]*If[t == 3, x, 1]] + If[d == 0, 0, b[u, d-1, If[t == 2, 3, 1]]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n+1, n+1, 1] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

Crossrefs

Cf. A097692.

Cf. A007318, A005773.

Cf. A168491, A099323, A001405, A005773, A001700, A026378, A005573, A122898.

Sequence in context: A188137 A201165 A171488 * A348451 A104710 A039598

Adjacent sequences: A171648 A171649 A171650 * A171652 A171653 A171654

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 14 2009

Extensions

Corrected by Philippe Deléham, Dec 18 2009

Status

approved