A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

1, 2, 11, 100, 1257, 20076, 387739, 8766248, 226739489, 6595646860, 212944033051, 7550600079672, 291527929539433, 12169325847587832, 545918747361417291, 26183626498897556176, 1336713063706757646465

Offset

1,2

Comments

A diagonal of Narayana array (A008550).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From N. J. A. Sloane, Oct 08 2012

Formula

From Vaclav Kotesovec, Apr 18 2014, extended Dec 01 2021: (Start)

a(n) = Hypergeometric2F1([1-n,-n], [2], -1+n).

a(n) ~ exp(2*sqrt(n)-2) * n^(n-7/4) / (2*sqrt(Pi)) * (1 + 119/(48*sqrt(n))). (End)

Maple

A099169:= n-> add( binomial(n, j)*binomial(n-1, j)*(n-1)^j/(j+1), j=0..n-1);
seq( A099169(n), n=1..30) # G. C. Greubel, Feb 16 2021

Mathematica

Join[{1}, Table[Sum[Binomial[n, k]Binomial[n, k+1](n-1)^k, {k, 0, n-1}]/n, {n, 2, 20}]] (* Harvey P. Dale, Oct 07 2013 *)
Table[Hypergeometric2F1[1-n, -n, 2, -1+n], {n, 1, 20}] (* Vaclav Kotesovec, Apr 18 2014 *)

Prog

(Sage)
def A099169(n): return sum( binomial(n, j)*binomial(n-1, j)*(n-1)^j/(j+1) for j in [0..n-1])
[A099169(n) for n in [1..30]] # G. C. Greubel, Feb 16 2021
(Magma)
A099169:= func< n | (&+[Binomial(n, j)*Binomial(n-1, j)*(n-1)^j/(j+1): j in [0..n-1]]) >;
[A099169(n): n in [1..30]]; // G. C. Greubel, Feb 16 2021
(PARI) a(n) = (1/n) * sum(k=0, n-1, binomial(n, k) * binomial(n, k+1) * (n-1)^k); \\ Michel Marcus, Feb 16 2021

Crossrefs

Cf. A092366, A187018, A187019, A187021.

Cf. A008550, A204057, A243631.

Sequence in context: A230889 A003579 A282640 * A143135 A205806 A220433

Adjacent sequences: A099166 A099167 A099168 * A099170 A099171 A099172

Keyword

nonn,easy

Author

Ralf Stephan, Oct 09 2004

Status

approved