A335183 - OEIS

%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