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

Author

Vladimir Kruchinin, Oct 06 2015

Keywords

• nonn
• easy

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

• 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.

Crossrefs

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

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

Cf. A088218, A163455, A165817.

Programs

Magma

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

Mathematica

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

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);

PARI

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

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

G.f.: 1/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025