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A179381 Row sums of A179318. 6
1, 2, 4, 10, 26, 78, 236, 770, 2520, 8606, 29364, 103302, 362226, 1298882, 4645670, 16897224, 61296686, 225457006, 826950080, 3067763394, 11353597198, 42414220022, 158095481910, 594108418428, 2227714454332, 8412269224862, 31704876569698, 120223392641084, 455053649594196, 1731861709709542, 6579658381972974 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..1650

StackExchange, Infinite product with the Catalan numbers, Mar 12 2018

FORMULA

G.f.: -1 + prod(n>=1, 1/(1-C(n-1)*x^n), where C(n) = A000108(n). - Vladimir Kruchinin, Aug 18 2014

a(n) = s(1,n), where s(m,n) = C(n-1)+sum(k=m..n/2, C(k-1)*s(k,n-k), a(n,n) = C(n-1), C(n) are the Catalan numbers (A000108). - Vladimir Kruchinin, Sep 06 2014

a(n) ~ c * 4^n / n^(3/2), where c = 1 / (4*sqrt(Pi) * Product_{k>=1} (1 - binomial(2*k-2,k-1) / (k * 4^k))) = 0.2422046382280667... - Vaclav Kotesovec, Mar 08 2018

EXAMPLE

The table has shape A000041 and begins:

1

1 1

2 1 1

5 2 1 1 1

14 5 2 2 1 1 1

so

a(n) begins 1 2 4 10 26 ...

PROG

(PARI)

N = 66;  x = 'x +O('x^N);

C(n) = binomial(2*n, n)/(n+1);

gf = -1 + 1/prod(n=1, N, 1 - C(n-1)*x^n );

Vec(gf)

\\ Joerg Arndt, Aug 18 2014

(Maxima)

C(n):=  1/(n+1)*binomial(2*n, n);

s(m, n):=if m>n then 0 else if n=m then C(n-1) else sum(C(k-1)*s(k, n-k), k, m, ceiling(n/2))+C(n-1);

makelist(s(1, n), n, 1, 27);  /* Vladimir Kruchinin, Sep 06 2014 */

CROSSREFS

Cf. A000108, A318264.

Sequence in context: A148101 A052854 A148102 * A096807 A003239 A195924

Adjacent sequences:  A179378 A179379 A179380 * A179382 A179383 A179384

KEYWORD

easy,nonn

AUTHOR

Alford Arnold, Jul 12 2010

EXTENSIONS

Terms 8606 and beyond (using Kruchinin's formula) by Joerg Arndt, Aug 18 2014

STATUS

approved

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Last modified November 28 20:08 EST 2020. Contains 338750 sequences. (Running on oeis4.)