A335183 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, 4, 4, 36, 60, 24, 288, 688, 560, 160, 2240, 7080, 8760, 5040, 1120, 17304, 68712, 114576, 99456, 44352, 8064, 133672, 642824, 1351840, 1572480, 1055040, 384384, 59136, 1034880, 5864640, 14912064, 21778560, 19536000, 10695168, 3294720, 439296
Offset: 0
Examples
Table T(n,k) (with rows n >= 0 and columns k = 0..n) begins as follows:
0;
4, 4;
36, 60, 24;
288, 688, 560, 160;
2240, 7080, 8760, 5040, 1120;
17304, 68712, 114576, 99456, 44352, 8064;
133672, 642824, 1351840, 1572480, 1055040, 384384, 59136;
...
Programs
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Mathematica
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, 7}, {l, 0, m}] // Flatten (* Jean-François Alcover, Jan 09 2014 from the original version of A126936 *) -
PARI
T(n,k) = sum(j=1, n, 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k)); tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n,k), ", "); ); print(); ); }
Formula
T(n,n) = A069722(n+1) for n >= 0.
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)))).
Comments