A187756 a(n) = n^2 * (4*n^2 - 1) / 3.

0, 1, 20, 105, 336, 825, 1716, 3185, 5440, 8721, 13300, 19481, 27600, 38025, 51156, 67425, 87296, 111265, 139860, 173641, 213200, 259161, 312180, 372945, 442176, 520625, 609076, 708345, 819280, 942761, 1079700, 1231041, 1397760, 1580865, 1781396, 2000425

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5.

a(n) = a(-n) for all n in Z.

a(n) = n * A000447(n).

G.f. A144853(x) = 1 / (1 - a(1)*x / (1 - a(2)*x / (1 - a(3)*x / ... ))).

From Amiram Eldar, Jun 21 2026: (Start)

Sum_{n>=1} 1/a(n) = 6 - Pi^2/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - Pi^2/4 - 6. (End)

Example

G.f. = x + 20*x^2 + 105*x^3 + 336*x^4 + 825*x^5 + 1716*x^6 + 3185*x^7 + ...

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 20, 105, 336}, 40] (* Harvey P. Dale, Mar 26 2016 *)
(* Alternative: *)
a[ n_] := SeriesCoefficient[ x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5, {x, 0, Abs[n]}]; (* Michael Somos, Dec 26 2016 *)

Prog

(PARI) {a(n) = polcoeff( x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5 + x * O(x^n), abs(n))};
(Maxima) A187756(n):=n^2*(4*n^2-1)/3$ makelist(A187756(n), n, 0, 20); /* Martin Ettl, Jan 07 2013 */
(Magma) [n^2*(4*n^2-1)/3: n in [0..50]]; // G. C. Greubel, Aug 10 2018

Crossrefs

Cf. A000447, A144853, A063496 (first differences).

Sequence in context: A063785 A181703 A334419 * A248087 A209547 A278642

Adjacent sequences: A187753 A187754 A187755 * A187757 A187758 A187759

Keyword

nonn,easy

Author

Michael Somos, Jan 03 2013

Status

approved