A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2*r+k,n)/(n+2*r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 20, 44, 0, 1, 8, 36, 120, 280, 0, 1, 10, 56, 236, 800, 1936, 0, 1, 12, 80, 400, 1656, 5696, 14128, 0, 1, 14, 108, 620, 2960, 12192, 42416, 107088, 0, 1, 16, 140, 904, 4840, 22592, 92960, 326304, 834912, 0, 1, 18, 176, 1260, 7440, 38352, 176800, 727824, 2572992, 6652608, 0

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.

G.f. of column k: B(x)^k where B(x) is the g.f. of A346626.

B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+2) for n > 0.

Example

Square array begins:
   1,     1,     1,     1,      1,      1,      1, ...
   0,     2,     4,     6,      8,     10,     12, ...
   0,     8,    20,    36,     56,     80,    108, ...
   0,    44,   120,   236,    400,    620,    904, ...
   0,   280,   800,  1656,   2960,   4840,   7440, ...
   0,  1936,  5696, 12192,  22592,  38352,  61248, ...
   0, 14128, 42416, 92960, 176800, 308560, 507152, ...

Prog

(PARI) T(n, k, t=1, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Crossrefs

Columns k=0..1 give A000007, A346626.

Cf. A033877, A071949, A378237, A378238, A378239, A378240.

Sequence in context: A329020 A351640 A173003 * A335461 A294411 A274390

Adjacent sequences: A378232 A378233 A378235 * A378237 A378238 A378239

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Nov 20 2024

Status

approved