A387914 a(n) = 3*binomial(2*n+4,n+3)/(n+4) - (n+2)*(n+3)/2 - 1.

2, 17, 74, 275, 972, 3395, 11888, 41934, 149159, 534809, 1931448, 7020299, 25662704, 94286983, 347993756, 1289624318, 4796857039, 17902146389, 67016296388, 251577049756, 946844533397, 3572042253827, 13505406670374, 51166197843500, 194214400833977, 738494264900601, 2812744285440500

Offset

1,1

Comments

Similar to A387913.

When n+1 >= 5 is prime, a(n) seems to be divisible by n+1.

Links

Georg Fischer, Table of n, a(n) for n = 1..500

Formula

a(n) = 3*binomial(2*n+4,n+3)/(n+4)-(n+2)*(n+3)/2-1.

a(n) = A000245(n+2)-A000124(n+2).

From Stefano Spezia, Sep 12 2025: (Start)

G.f.: 1 - (4 - 5*x + 2*x^2)/(1 - x)^3 + (1 - sqrt(1 - 4*x) + (sqrt(1 - 4*x) - 3)*x - 2*x^3)/(2*x^4).

E.g.f.: 1 - exp(x)*(8 + 6*x + x^2)/2 - 6*exp(2*x)*(1 - x)*BesselI(0, 2*x)/x^2 + 6*exp(2*x)*(1 - x + x^2)*BesselI(1, 2*x)/x^3. (End)

D-finite with recurrence (n+4)*a(n) +3*(-2*n-5)*a(n-1) +9*(n+1)*a(n-2) +2*(-2*n+1)*a(n-3) +3*(-4*n-1)=0. - R. J. Mathar, Sep 26 2025

Example

The value a(3) = 74 is the Hodge number h^{1,2} of a Calabi-Yau hypersurface in a 4-dimensional toric variety.

Prog

(SageMath) [3*binomial(2*n+4, n+3)/(n+4)-(n+2)*(n+3)/2-1 for n in range(1, 33)]

Crossrefs

Cf. A387913, A000245, A000124.

Sequence in context: A054568 A338087 A235471 * A297411 A183175 A060352

Adjacent sequences: A387911 A387912 A387913 * A387915 A387916 A387917

Keyword

nonn,easy

Author

F. Chapoton, Sep 12 2025

Extensions

a(19) and following terms corrected by Georg Fischer, Sep 16 2025

Status

approved