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A270447 Binomial transform(2) of Catalan numbers. 5
1, 3, 11, 43, 174, 721, 3044, 13059, 56837, 250690, 1119612, 5059561, 23119628, 106753404, 497762380, 2342096579, 11113027686, 53138757319, 255892224332, 1240217043450, 6046131132030, 29631889507380, 145923474439800, 721733515299225, 3583733352377724 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = Sum_{k=0..n} (T(n,k)*C(k)), where C(k) is Catalan numbers (A000108), T(n,k) - triangle of A092392.

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

G.f.: C(C(x))*(1-C(x))^2/(((1-C(x))^2)-x)/x, where C(x)=(1-sqrt(1-4*x))/2.

Recurrence: 3*(n-1)*n*(n+1)*(2*n - 3)*a(n) = 16*(n-1)*n*(5*n^2 - 10*n + 3)*a(n-1) - 16*(n-1)*(2*n - 1)*(11*n^2 - 33*n + 24)*a(n-2) + 8*(2*n - 3)*(2*n - 1)*(4*n - 9)*(4*n - 7)*a(n-3). - Vaclav Kotesovec, Mar 17 2016

a(n) ~ 2^(4*n + 1/2) / (sqrt(Pi) * 3^(n - 1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2016

a(n) = [x^n] (1 - sqrt(1 - 4*x))/(2*x*(1 - x)^(n+1)). - Ilya Gutkovskiy, Nov 01 2017

MATHEMATICA

Table[Sum[Binomial[2*k, k]/(k+1) * Binomial[2*n-k, n], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 17 2016 *)

PROG

(Maxima)

a(n):=sum((binomial(2*k, k)*binomial(2*n-k, n))/(k+1), k, 0, n);

(PARI) a(n) = sum(i=0, n, (binomial(2*i, i)*binomial(2*n-i, n))/(i+1)); \\ Altug Alkan, Mar 17 2016

CROSSREFS

Cf. A000108, A007317, A092392.

Sequence in context: A007583 A026671 A026876 * A151090 A059278 A151091

Adjacent sequences:  A270444 A270445 A270446 * A270448 A270449 A270450

KEYWORD

nonn

AUTHOR

Vladimir Kruchinin, Mar 17 2016

STATUS

approved

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Last modified November 18 15:55 EST 2018. Contains 317323 sequences. (Running on oeis4.)