The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A082181 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A082181

a(0) = 1, for n>=1, a(n) = Sum_{k=0..n} 9^k*N(n,k), where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
(history; published version)

#51 by Michel Marcus at Tue May 24 04:59:28 EDT 2022

STATUS

reviewed

approved

#50 by Joerg Arndt at Tue May 24 04:43:25 EDT 2022

STATUS

proposed

reviewed

#49 by Michel Marcus at Tue May 24 02:31:52 EDT 2022

STATUS

editing

proposed

#48 by Michel Marcus at Tue May 24 02:31:49 EDT 2022

COMMENTS

From Gary W. Adamson, Jul 08 2011: (Start)

...

- Gary W. Adamson, Jul 08 2011

... (End)

STATUS

proposed

editing

#47 by Joerg Arndt at Tue May 24 02:30:11 EDT 2022

STATUS

editing

proposed

#46 by Joerg Arndt at Tue May 24 02:29:38 EDT 2022

COMMENTS

- _Gary W. Adamson, _, Jul 08 2011

STATUS

proposed

editing

#45 by Michel Marcus at Tue May 24 01:20:31 EDT 2022

STATUS

editing

proposed

#44 by Michel Marcus at Tue May 24 01:20:22 EDT 2022

COMMENTS

More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) are given by: a(n) = sum(Sum_{k=0..n, } (m+1)^k*N(n,k)).

STATUS

proposed

editing

#43 by G. C. Greubel at Mon May 23 23:42:08 EDT 2022

STATUS

editing

proposed

#42 by G. C. Greubel at Mon May 23 23:41:40 EDT 2022

NAME

a(0) = 1; , for n>=1, a(n) = sum(Sum_{k=0..n, } 9^k*N(n,k)), , where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

COMMENTS

1, 1

9, 9, 9

1, 1, 1, 1

9, 9, 9, 9, 9

1, 1, 1, 1, 1, 1

...

FORMULA

a(n) = (10*(2n2*n-1)*a(n-1) - 64*(n-2)*a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe Deléham, Aug 19 2005

PROG

(Magma) [(&+[Binomial(n, k)*Binomial(n-1, k)*9^k/(k+1): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2022

(SageMath) [sum(binomial(n, k)*binomial(n-1, k)*9^k/(k+1) for k in (0..n)) for n in (0..30)] # G. C. Greubel, May 23 2022

CROSSREFS

Cf. A001003, A001263, A007564, A059231, A088617.

STATUS

approved

editing