A229147 a(n) = n^4*(3*n+2).

0, 5, 128, 891, 3584, 10625, 25920, 55223, 106496, 190269, 320000, 512435, 787968, 1171001, 1690304, 2379375, 3276800, 4426613, 5878656, 7688939, 9920000, 12641265, 15929408, 19868711, 24551424, 30078125, 36558080, 44109603, 52860416, 62948009, 74520000

Offset

0,2

Comments

Number of ascending runs in {1,...,n}^5.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

G.f.: (x^4+58*x^3+198*x^2+98*x+5)*x/(x-1)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), with a(0)=0, a(1)=5, a(2)=128, a(3)=891, a(4)=3584, a(5)=10625. - Harvey P. Dale, Aug 14 2015

E.g.f.: exp(x)*x*(5 + 59*x + 87*x^2 + 32*x^3 + 3*x^4). - Stefano Spezia, Jul 17 2024

From Amiram Eldar, Nov 17 2025: (Start)

Sum_{n>=1} 1/a(n) = Pi^4/180 + 3*Pi^2/16 - 9*sqrt(3)*Pi/32 - 3*zeta(3)/4 + 81*log(3)/32 - 81/32.

Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/1440 + 3*Pi^2/32 - 9*sqrt(3)*Pi/16 - 9*zeta(3)/16 + 81/32. (End)

Maple

a:= n-> n^4*(3*n+2):
seq(a(n), n=0..40);

Mathematica

Table[n^4 (3n+2), {n, 0, 30}] (* Harvey P. Dale, Aug 14 2015 *)
(* Alternative: *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 5, 128, 891, 3584, 10625}, 40] (* Harvey P. Dale, Aug 14 2015 *)

Crossrefs

Row n=5 of A229079.

Cf. A000583, A016789.

Sequence in context: A157438 A142803 A208859 * A224250 A316986 A355085

Adjacent sequences: A229144 A229145 A229146 * A229148 A229149 A229150

Keyword

nonn,easy,changed

Author

Alois P. Heinz, Sep 15 2013

Status

approved