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

0,2

Comments

This was the original version of A126936.

Links

Table of n, a(n) for n=0..35.

Formula

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

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

T(n,0) = A126936(n,0) - binomial(2*n, n) = A067001(n,n) - A000984(n) 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)))).

Example

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;
  ...

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, 11}, {l, 0, m}] // Flatten (* Jean-François Alcover, Jan 09 2014_ from the original version of A126936 *)

Prog

(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(); ); }

Crossrefs

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

Sequence in context: A249807 A180064 A105350 * A129357 A100303 A192348

Adjacent sequences: A335180 A335181 A335182 * A335184 A335185 A335186

Keyword

nonn,tabl

Author

Petros Hadjicostas, May 25 2020

Status

approved