A076038 - OEIS

A076038

Square array read by ascending antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 10, 14, 14, 1, 5, 17, 35, 42, 42, 1, 6, 26, 74, 126, 132, 132, 1, 7, 37, 137, 326, 462, 429, 429, 1, 8, 50, 230, 726, 1446, 1716, 1430, 1430, 1, 9, 65, 359, 1434, 3858, 6441, 6435, 4862, 4862, 1, 10, 82, 530, 2582, 8952, 20532, 28770, 24310, 16796, 16796

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OFFSET

0,5

LINKS

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

FORMULA

A(n, m) = 1/(m+1)*Sum_{k=0..m} binomial(2*m-k, m)*(k+1)*(n-m)^k, m=0..n.

EXAMPLE

Array begins as:

1 1 2 5 14 42 ... (n=0)

1 2 5 14 42 132 ... (n=1)

1 3 10 35 126 ... (n=2)

1 4 17 74 326 ...

...

MATHEMATICA

Unprotect[Power]; Power[0, 0]=1; Protect[Power]; A[n_, m_]:= 1/(m+1)*Sum[Binomial[2*m-k, m]*(k+1)*(n-m)^k, {k, 0, m}]; Table[A[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* Stefano Spezia, Sep 01 2025 *)

CROSSREFS

Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076037, A067347.

Cf. A059718.

Sequence in context: A064580 A009766 A059718 * A095788 A071944 A309495

Adjacent sequences: A076035 A076036 A076037 * A076039 A076040 A076041

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Oct 29 2002

EXTENSIONS

More terms from Vladeta Jovovic, Jul 18 2003

a(63)-a(65) from Stefano Spezia, Sep 01 2025

STATUS

approved