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A378237
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+3*r+k,n)/(n+3*r+k) for k > 0.
5
1, 1, 0, 1, 2, 0, 1, 4, 10, 0, 1, 6, 24, 74, 0, 1, 8, 42, 188, 642, 0, 1, 10, 64, 350, 1680, 6082, 0, 1, 12, 90, 568, 3234, 16212, 60970, 0, 1, 14, 120, 850, 5440, 31878, 164584, 635818, 0, 1, 16, 154, 1204, 8450, 54888, 328426, 1732172, 6826690, 0, 1, 18, 192, 1638, 12432, 87402, 574848, 3494142, 18728352, 74958914, 0
OFFSET
0,5
FORMULA
G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A349310.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + 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, 10, 24, 42, 64, 90, 120, ...
0, 74, 188, 350, 568, 850, 1204, ...
0, 642, 1680, 3234, 5440, 8450, 12432, ...
0, 6082, 16212, 31878, 54888, 87402, 131964, ...
0, 60970, 164584, 328426, 574848, 931770, 1433544, ...
PROG
(PARI) T(n, k, t=1, u=3) = 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, A349310.
Sequence in context: A274390 A244128 A016584 * A293961 A378239 A378238
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 20 2024
STATUS
approved