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

%I A096164 #25 Dec 21 2019 15:08:47
%S A096164 1,2,1,5,4,1,12,14,6,1,30,44,27,8,1,74,133,104,44,10,1,185,388,369,
%T A096164 200,65,12,1,460,1110,1236,814,340,90,14,1,1150,3120,3980,3072,1560,
%U A096164 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
%N A096164 Triangle T(n,m) read by rows: matrix product A053121 * A038207.
%C A096164 Building the product with the two matrices swapped generates A039598 = A038207 * A053121.
%F A096164 Equals A053121 * A038207 = A053121 * (A007318)^2.
%F A096164 T(n,m) = Sum_{k=0..n} A053121(n,k)* A038207(k,m).
%F A096164 T(n,n-2) = A014106(n-1).
%F A096164 Conjecture: Sum_{m=0..n} T(n,m) = A126931(n). - _R. J. Mathar_, Mar 25 2010
%F A096164 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
%F A096164 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
%e A096164 The triangle starts
%e A096164    1;
%e A096164    2,  1;
%e A096164    5,  4,  1;
%e A096164   12, 14,  6,  1;
%e A096164   30, 44, 27,  8,  1;
%p A096164 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:
%p A096164 A038207 := proc(i,j) if j> i then 0; else binomial(i,j)*2^(i-j) ; end if; end proc:
%p A096164 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) ;
%t A096164 a53121[n_, k_] /; n < k || OddQ[n - k] = 0;
%t A096164 a53121[n_, k_] := (k + 1) Binomial[n + 1, (n - k)/2]/(n + 1);
%t A096164 a38207[n_, k_] := Sum[Binomial[n, i] Binomial[i, k], {i, 0, n}];
%t A096164 T[n_, k_] := Sum[a53121[n, j] a38207[j, k], {j, 0, n}];
%t A096164 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 21 2019 *)
%o A096164 (Maxima)
%o A096164 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 */
%Y A096164 Cf. A039598, A007318, A038207, A053121, A001700, A049310.
%K A096164 nonn,tabl
%O A096164 1,2
%A A096164 _Gary W. Adamson_, Jun 19 2004
%E A096164 Edited, definition rewritten, program and more terms added by _R. J. Mathar_, Mar 25 2010