login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

T(n,k) = Sum_{j=1..n} 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k), triangle read by rows (n >= 0 and 0 <= k <= n).
0

%I #26 May 27 2020 10:25:41

%S 0,4,4,36,60,24,288,688,560,160,2240,7080,8760,5040,1120,17304,68712,

%T 114576,99456,44352,8064,133672,642824,1351840,1572480,1055040,384384,

%U 59136,1034880,5864640,14912064,21778560,19536000,10695168,3294720,439296

%N T(n,k) = Sum_{j=1..n} 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k), triangle read by rows (n >= 0 and 0 <= k <= n).

%C This was the original version of A126936.

%F T(n,n) = A069722(n+1) for n >= 0.

%F T(n,k) = A126936(n,k) = A067001(n,n-k) for n >= k >= 1.

%F T(n,0) = A126936(n,0) - binomial(2*n, n) = A067001(n,n) - A000984(n) for n >= 0.

%F Bivariate o.g.f.: Sum_{n,k >= 0} T(n,k)*x^n*y^k = -1/sqrt(1 - 4*x) + sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))).

%e Table T(n,k) (with rows n >= 0 and columns k = 0..n) begins as follows:

%e 0;

%e 4, 4;

%e 36, 60, 24;

%e 288, 688, 560, 160;

%e 2240, 7080, 8760, 5040, 1120;

%e 17304, 68712, 114576, 99456, 44352, 8064;

%e 133672, 642824, 1351840, 1572480, 1055040, 384384, 59136;

%e ...

%t t[l_, m_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, 1, m}]; Table[t[l, m], {m, 0, 11}, {l, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 09 2014_ from the original version of A126936 *)

%o (PARI) T(n,k) = sum(j=1, n, 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k));

%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n,k), ", "); ); print(); ); }

%Y Cf. A000984, A067001, A069722 (main diagonal), A126936.

%K nonn,tabl

%O 0,2

%A _Petros Hadjicostas_, May 25 2020