The On-Line Encyclopedia of Integer Sequences (OEIS)

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A161797

G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3).
(history; published version)

#10 by Vaclav Kotesovec at Fri Nov 05 13:10:20 EDT 2021

STATUS

proposed

approved

#9 by Seiichi Manyama at Fri Nov 05 13:05:49 EDT 2021

STATUS

editing

proposed

#8 by Seiichi Manyama at Fri Nov 05 13:02:40 EDT 2021

LINKS

Seiichi Manyama, <a href="/A161797/b161797.txt">Table of n, a(n) for n = 0..1000</a>

#7 by Seiichi Manyama at Fri Nov 05 13:00:42 EDT 2021

CROSSREFS

Cf. A109081, A321798, A321799.

STATUS

approved

editing

#6 by Vaclav Kotesovec at Sat Nov 18 10:57:16 EST 2017

STATUS

editing

approved

#5 by Vaclav Kotesovec at Sat Nov 18 10:57:09 EST 2017

FORMULA

a(n) ~ sqrt(sqrt((159 + 100*sqrt(3))/13) - 2 - 5/sqrt(3)) * (3 + 2*sqrt(3) + sqrt(153 + 100*sqrt(3))/3)^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 5/2)). - Vaclav Kotesovec, Nov 18 2017

#4 by Vaclav Kotesovec at Sat Nov 18 08:32:19 EST 2017

FORMULA

Recurrence: 3*(n+1)*(3*n - 2)*(3*n + 2)*(2145*n^4 - 14355*n^3 + 33844*n^2 - 32668*n + 10380)*a(n) = 3*(115830*n^7 - 833085*n^6 + 2195691*n^5 - 2521863*n^4 + 998671*n^3 + 259048*n^2 - 263292*n + 41520)*a(n-1) + 3*(n-2)*(19305*n^6 - 129195*n^5 + 315651*n^4 - 367201*n^3 + 219176*n^2 - 66584*n + 7608)*a(n-2) + 3*(n-3)*(n-2)*(64350*n^5 - 334125*n^4 + 546005*n^3 - 255608*n^2 - 71320*n + 54328)*a(n-3) - 23*(n-4)*(n-3)*(n-2)*(2145*n^4 - 5775*n^3 + 3649*n^2 + 535*n - 654)*a(n-4). - Vaclav Kotesovec, Nov 18 2017

#3 by Vaclav Kotesovec at Sat Nov 18 08:30:17 EST 2017

MATHEMATICA

Table[Sum[Binomial[n, k]/(n-k+1) * Binomial[n+2*k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Nov 18 2017 *)

STATUS

approved

editing

#2 by Russ Cox at Fri Mar 30 18:37:17 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jun 19 2009

Discussion

Fri Mar 30

18:37

OEIS Server: https://oeis.org/edit/global/213

#1 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010

NAME

G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3).

DATA

1, 1, 4, 16, 71, 336, 1660, 8464, 44207, 235306, 1271807, 6961307, 38508659, 214950425, 1209170536, 6848080767, 39014400171, 223439516338, 1285660965508, 7428738358924, 43087099589998, 250766507928988, 1464026402082801

OFFSET

0,3

FORMULA

a(n) = Sum_{k=0..n} C(n,k)/(n-k+1) * C(n+2*k-1,n-k).

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then

a(n,m) = Sum_{k=0..n} C(n+m-1,k)*m/(n-k+m) * C(n+2*k-1,n-k).

G.f.: A(x) = (1/x)*serreverse[x/(1 + x/(1 - x)^3)].

PROG

(PARI) {a(n, m=1)=sum(k=0, n, binomial(n+m-1, k)*m/(n-k+m)*binomial(n+2*k-1, n-k))}

CROSSREFS

Cf. A109081.

KEYWORD

nonn

AUTHOR

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 19 2009

STATUS

approved