A262977 a(n) = binomial(4*n-1,n).

1, 3, 21, 165, 1365, 11628, 100947, 888030, 7888725, 70607460, 635745396, 5752004349, 52251400851, 476260169700, 4353548972850, 39895566894540, 366395202809685, 3371363686069236, 31074067324187580, 286845713747883300, 2651487106659130740, 24539426037817994160

Offset

0,2

Comments

From Gus Wiseman, Sep 28 2022: (Start)

Also the number of integer compositions of 4n with alternating sum 2n, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A348614. The a(12) = 21 compositions are:

(6,2) (1,2,5) (1,1,5,1) (1,1,1,1,4)

(2,2,4) (2,1,4,1) (1,1,2,1,3)

(3,2,3) (3,1,3,1) (1,1,3,1,2)

(4,2,2) (4,1,2,1) (1,1,4,1,1)

(5,2,1) (5,1,1,1) (2,1,1,1,3)

(2,1,2,1,2)

(2,1,3,1,1)

(3,1,1,1,2)

(3,1,2,1,1)

(4,1,1,1,1)

The following pertain to this interpretation:

- The case of partitions is A000712, reverse A006330.

- Allowing any alternating sum gives A013777 (compositions of 4n).

- A011782 counts compositions of n.

- A034871 counts compositions of 2n with alternating sum 2k.

- A097805 counts compositions by alternating (or reverse-alternating) sum.

- A103919 counts partitions by sum and alternating sum (reverse: A344612).

- A345197 counts compositions by length and alternating sum.

(End)

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

a(n) = A071919(3n-1,n+1) = A097805(4n,n+1). - Gus Wiseman, Sep 28 2022

From Peter Bala, Feb 14 2024: (Start)

a(n) = (-1)^n * binomial(-3*n, n).

a(n) = hypergeom([1 - 3*n, -n], [1], 1).

The g.f. A(x) satisfies A(x/(1 + x)^4) = 1/(1 - 3*x). (End)

a(n) = Sum_{k = 0..n} binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). - Peter Bala, Sep 16 2024

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.

Cf. A000346, A000984, A001700, A002458, A025047, A058622, A081294, A294175.

Sequence in context: A118353 A365132 A371817 * A361375 A371771 A214391

Adjacent sequences: A262974 A262975 A262976 * A262978 A262979 A262980

Keyword

nonn,easy,changed

Author

Vladimir Kruchinin, Oct 06 2015

Status

approved