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A378239
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(2*n+2*r+k,n)/(2*n+2*r+k) for k > 0.
5
1, 1, 0, 1, 2, 0, 1, 4, 12, 0, 1, 6, 28, 100, 0, 1, 8, 48, 248, 968, 0, 1, 10, 72, 452, 2480, 10208, 0, 1, 12, 100, 720, 4680, 26688, 113792, 0, 1, 14, 132, 1060, 7728, 51504, 301648, 1318832, 0, 1, 16, 168, 1480, 11800, 87104, 591312, 3531424, 15732064, 0
OFFSET
0,5
FORMULA
G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/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 A219534.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-1,k+3) for n > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, 12, ...
0, 12, 28, 48, 72, 100, 132, ...
0, 100, 248, 452, 720, 1060, 1480, ...
0, 968, 2480, 4680, 7728, 11800, 17088, ...
0, 10208, 26688, 51504, 87104, 136352, 202560, ...
0, 113792, 301648, 591312, 1017184, 1621280, 2454256, ...
PROG
(PARI) T(n, k, t=2, 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
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 20 2024 based on suggestions from Mikhail Kurkov.
STATUS
approved