A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 19, 9, 5, 1, 1, 51, 26, 11, 6, 1, 1, 141, 70, 34, 13, 7, 1, 1, 393, 197, 92, 43, 15, 8, 1, 1, 1107, 553, 265, 117, 53, 17, 9, 1, 1, 3139, 1570, 751, 346, 145, 64, 19, 10, 1, 1, 8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1

Offset: 0

Author

Gary W. Adamson & Roger L. Bagula, Mar 26 2009

Keywords

• nonn
• tabl

Comments

Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A002426 and where g(x) is the g.f. of A005043. - Philippe Deléham, Dec 05 2009

Matrix product P * Q * P^(-1), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158815 and A171243. - Peter Bala, Jul 13 2021

Examples

			First rows of the triangle:
     1;
     1,    1;
     3,    1,    1;
     7,    4,    1,   1;
    19,    9,    5,   1,   1;
    51,   26,   11,   6,   1,   1;
   141,   70,   34,  13,   7,   1,  1;
   393,  197,   92,  43,  15,   8,  1,  1;
  1107,  553,  265, 117,  53,  17,  9,  1,  1;
  3139, 1570,  751, 346, 145,  64, 19, 10,  1, 1;
  8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1;
		

Crossrefs

T(n, 0) = A002426(n), A005773 (row sums).

Cf. A046899, A007318, A158815, A171243.

Programs

Maple

A158793 := proc (n, k)
  add((-1)^(n+j)*binomial(n, j)*binomial(2*j-k, j-k), j = k..n);
end proc:
seq(seq(A158793(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

Mathematica

T[n_, k_] := (-1)^(k + n) Binomial[n, k] HypergeometricPFQ[{k/2 + 1/2, k/2 + 1, k - n}, {k + 1, k + 1}, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Jul 17 2021 *)

Formula

T(n, m) = Sum_{k=m..n-1} A130595(n,k) * A092392(k+1,m+1), with the triangular interpretation of A092392.

Conjecture: T(n,1) = A113682(n-1). - R. J. Mathar, Oct 06 2009

Sum_{k=0..n} T(n,k)*x^k = A002426(n), A005773(n+1), A000244(n), A126932(n) for x = 0,1,2,3 respectively. - Philippe Deléham, Dec 03 2009

T(n, k) = (-1)^(k + n) binomial(n, k) hypergeom([k/2 + 1/2, k/2 + 1, k - n], [k + 1, k + 1], 4). - Peter Luschny, Jul 17 2021

Extensions

Simplified definition from R. J. Mathar, Oct 06 2009