A236194 a(n) = binomial(3n+1, n-1).

1, 7, 45, 286, 1820, 11628, 74613, 480700, 3108105, 20160075, 131128140, 854992152, 5586853480, 36576848168, 239877544005, 1575580702584, 10363194502115, 68248282427325, 449972009097765, 2969831763694950, 19619725782651120, 129728497393775280, 858478958817125100

Offset

1,2

Comments

This sequence is related to A006013 by a(n)/n = A006013(n)/2.

Links

Bruno Berselli, Table of n, a(n) for n = 1..100

Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Hua Xin, Lattice points of flow polytopes related to caracol graphs, AIMS Elect. Res. Archive 33(10) (2025) 6141-6175. See p. 6160, Table 5.

Formula

G.f.: (sqrt(4-27*x)*cos((2/3)*arcsin((3/2)*sqrt(3*x))) + sqrt(3*x)*sin((2/3)*arcsin((3/2)*sqrt(3*x))) - sqrt(4-27*x))/(3*sqrt(4-27*x)*x^2). - Emanuele Munarini, Oct 14 2014

From Peter Bala, Nov 04 2015: (Start)

With offset 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.

More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)

a(n) = [x^n] x/(1 - x)^(2*n+3). - Ilya Gutkovskiy, Oct 10 2017

From Karol A. Penson, Mar 02 2024: (Start)

G.f.: ((sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) - 12*i*sqrt(3)*sqrt(x))^(2/3) + (-sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) + 12*i*sqrt(3)*sqrt(x))^(2/3) - 8*sqrt(4 - 27*x))/(24*sqrt(4 - 27*x)*x^2), where i is the imaginary unit, i=sqrt(-1).

G.f.: hypergeometric3F2([5/3,2,7/3],[5/2,3],27*x/4).

G.f. = G satisfies the algebraic equation: 1 + (7*z-1)*G + (27*z-4)*z^2*G^2 + (27*z-4)*z^4*G^3 = 0. (End)

a(n) ~ 3^(3*n+3/2) / (2^(2*n+3) * sqrt(Pi*n)). - Amiram Eldar, Sep 05 2025

D-finite with recurrence -2*(n-1)*(2*n+1)*(n+1)*a(n) +3*n*(3*n-1)*(3*n+1)*a(n-1)=0. - R. J. Mathar, Mar 14 2026

Mathematica

Table[Binomial[3n+1, n-1], {n, 30}]

Prog

(Magma) [Binomial(3*n+1, n-1): n in [1..30]];
(Maxima) makelist(binomial(3*n+4, n), n, 0, 40); /* Emanuele Munarini, Oct 14 2014 */
(PARI) vector(30, n, binomial(3*n+1, n-1)) \\ Altug Alkan, Nov 04 2015
(SageMath) [binomial(3*n+1, n-1) for n in range(1, 31)] # G. C. Greubel, Nov 09 2022

Crossrefs

Cf. A006013; A025174: C(3n-1, n-1); A117671: C(3n+1, n+1).

Second column of the triangle A159841.

Third column of the triangle A119301.

Cf. A001764, A004319, A005809, A013698, A045721, A165817.

Cf. A045721, A117671.

Sequence in context: A342568 A386941 A323140 * A230760 A198629 A190973

Adjacent sequences: A236191 A236192 A236193 * A236195 A236196 A236197

Keyword

nonn,easy

Author

Bruno Berselli, Jan 20 2014

Status

approved