A387913 Hodge number h^{1,n-1} of a smooth hypersurface of degree n+2 in P^{n+1}.

1, 19, 101, 426, 1667, 6371, 24229, 92278, 352595, 1351934, 5200131, 20058104, 77558535, 300539939, 1166802821, 4537567326, 17672631539, 68923264010, 269128936779, 1052049481376, 4116715363271, 16123801840974, 63205303218251

Offset

1,2

Comments

This is the Hodge number h^{1, n-1} of a smooth hypersurface of degree n+2 in P^{n+1}, except for n=2 where the full Hodge number is 20. In this special case only, the Hodge number h^{1, n-1} sits at the center of the Hodge diamond, which explains the difference of 1.

Links

Table of n, a(n) for n=1..23.

G. Bini and A. Garbagnati, Quotients of the Dwork pencil, Journal of Geometry and Physics, 75 (2014), 173-198. Cf. Remark 2.3.

Formula

a(n) = binomial(2*n+3,n+2)-(n+2)^2.

From Stefano Spezia, Sep 12 2025: (Start)

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

E.g.f.: 1 - exp(x)*(4 + 5*x + x^2) + exp(2*x)*(4*x*BesselI(0, 2*x) - BesselI(1, 2*x) + 4*x*BesselI(1, 2*x))/x. (End)

Example

For n=1, an elliptic curve has Hodge number h^{1,0}=1. For n=2, a K3 surface has Hodge number h^{1,1}=20=19+1. For n=3, a smooth quintic threefold has Hodge number h^{1,2}=101.

Prog

(SageMath) [binomial(2*n+3, n+2)-(n+2)**2 for n in range(1, 26)]

Crossrefs

Cf. A000290, A001700, A387914.

Sequence in context: A287303 A142216 A027005 * A061696 A062643 A245528

Adjacent sequences: A387910 A387911 A387912 * A387914 A387915 A387916

Keyword

nonn,easy

Author

F. Chapoton, Sep 12 2025

Status

approved