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A262977 a(n) = binomial(4*n-1,n). 4
1, 3, 21, 165, 1365, 11628, 100947, 888030, 7888725, 70607460, 635745396, 5752004349, 52251400851, 476260169700, 4353548972850, 39895566894540, 366395202809685, 3371363686069236, 31074067324187580, 286845713747883300, 2651487106659130740, 24539426037817994160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

FORMULA

G.f.: A(x)=x*B'(x)/B(x), where B(x) if g.f. of A006632.

a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(3*n,k)).

a(n) = 3*A224274(n), for n > 0. - Michel Marcus, Oct 12 2015

From Peter Bala, Nov 04 2015: (Start)

The o.g.f. equals f(x)/g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A005810 (k = 0), A052203 (k = 1), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)

a(n) = [x^n] 1/(1 - x)^(3*n). - Ilya Gutkovskiy, Oct 03 2017

MATHEMATICA

Table[Binomial[4 n - 1, n], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2015 *)

PROG

(Maxima)

B(x):=sum(binomial(4*n-1, n-1)*3/(4*n-1)*x^n, n, 1, 30);

taylor(x*diff(B(x), x, 1)/B(x), x, 0, 20);

(MAGMA) [Binomial(4*n-1, n): n in [0..20]]; // Vincenzo Librandi, Oct 06 2015 *)

(PARI) a(n) = binomial(4*n-1, n); \\ Michel Marcus, Oct 06 2015

CROSSREFS

Cf. A006632, A002293, A004331, A005810, A052203, A224274, A257633.

Sequence in context: A058194 A179815 A118353 * A214391 A046637 A220103

Adjacent sequences:  A262974 A262975 A262976 * A262978 A262979 A262980

KEYWORD

nonn,easy

AUTHOR

Vladimir Kruchinin, Oct 06 2015

STATUS

approved

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Last modified March 18 19:58 EDT 2019. Contains 321293 sequences. (Running on oeis4.)