A096164 Triangle T(n,m) read by rows: matrix product A053121 * A038207.

1, 2, 1, 5, 4, 1, 12, 14, 6, 1, 30, 44, 27, 8, 1, 74, 133, 104, 44, 10, 1, 185, 388, 369, 200, 65, 12, 1, 460, 1110, 1236, 814, 340, 90, 14, 1, 1150, 3120, 3980, 3072, 1560, 532, 119, 16, 1, 2868, 8666, 12432, 10984, 6542, 2715, 784, 152, 18, 1, 7170, 23816, 37938, 37688, 25695, 12516, 4403, 1104, 189, 20, 1

Offset: 1

Author

Gary W. Adamson, Jun 19 2004

Keywords

• nonn
• tabl

Comments

Building the product with the two matrices swapped generates A039598 = A038207 * A053121.

Examples

			The triangle starts
   1;
   2,  1;
   5,  4,  1;
  12, 14,  6,  1;
  30, 44, 27,  8,  1;
		

Crossrefs

Cf. A039598, A007318, A038207, A053121, A001700, A049310.

Programs

Maple

A053121 := proc(n,k) if n< k or type(n-k,'odd') then 0; else (k+1)*binomial(n+1,(n-k)/2)/(n+1) ; end if; end proc:
A038207 := proc(i,j) if j> i then 0; else binomial(i,j)*2^(i-j) ; end if; end proc:
A096164 := proc(n,m) add( A053121(n,k)*A038207(k,m), k=0..n) ; end proc: seq(seq(A096164(n,m),m=0..n),n=0..15) ;

Mathematica

a53121[n_, k_] /; n < k || OddQ[n - k] = 0;
a53121[n_, k_] := (k + 1) Binomial[n + 1, (n - k)/2]/(n + 1);
a38207[n_, k_] := Sum[Binomial[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := Sum[a53121[n, j] a38207[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 21 2019 *)

Maxima

T(n,k):=(2*k*sum(binomial(j,-n-3*k+2*j)*binomial(n+k,j),j,0,n+k))/(n+k); /* Vladimir Kruchinin, Oct 12 2011 */

Formula

Equals A053121 * A038207 = A053121 * (A007318)^2.

T(n,m) = Sum_{k=0..n} A053121(n,k)* A038207(k,m).

T(n,n-2) = A014106(n-1).

Conjecture: Sum_{m=0..n} T(n,m) = A126931(n). - R. J. Mathar, Mar 25 2010

T(n,k) = (2*k*sum(j=0..n+k, binomial(j,-n-3*k+2*j)*binomial(n+k,j)))/(n+k). - Vladimir Kruchinin, Oct 12 2011

T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-2)^i. - Philippe Deléham, Feb 23 2012

Extensions

Edited, definition rewritten, program and more terms added by R. J. Mathar, Mar 25 2010