A060493 A diagonal of A036969.

0, 1, 21, 147, 627, 2002, 5278, 12138, 25194, 48279, 86779, 148005, 241605, 380016, 578956, 857956, 1240932, 1756797, 2440113, 3331783, 4479783, 5939934, 7776714, 10064110, 12886510, 16339635, 20531511, 25583481, 31631257, 38826012, 47335512, 57345288, 69059848

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See p. 5.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

From Benoit Cloitre, Mar 20 2004: (Start)

a(n) = n*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3)*(5*n - 1)/360.

a(n) = Sum_{k=1..n} k^2 * Sum_{i=1..k} i^2.

a(n) = Sum_{k=1..n} k^2*A000330(k). (End)

G.f.: -x*(4*x^3+21*x^2+14*x+1) / (x-1)^7. - Colin Barker, Dec 19 2012

a(n) = (2/(2*n)!) * Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+4) * binomial(2*n, n-j). - Peter Bala, Mar 31 2025

Sum_{n>=1} 1/a(n) = (5/1309) * (3750*sqrt(5)*arctanh(1/sqrt(5)) + 50688*log(2) + 9375*log(5) - 37766 - 750*sqrt(5*(5 + 2*sqrt(5)))*Pi). - Amiram Eldar, Dec 30 2025

E.g.f.: exp(x)*x*(360 + 3420*x + 5220*x^2 + 2415*x^3 + 396*x^4 + 20*x^5)/360. - Stefano Spezia, Apr 09 2026

Mathematica

a[n_] := n*(n+1)*(n+2)*(2*n+1)*(2*n+3)*(5*n-1)/360; Array[a, 30, 0] (* Amiram Eldar, Dec 30 2025 *)

Prog

(PARI) a(n)=n*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3)*(5*n - 1)/360
(Python)
def A060493(n): return n*(n*(n*(n*(n*(20*n+96)+155)+90)+5)-6)//360 # Chai Wah Wu, Apr 09 2026

Crossrefs

Cf. A000330, A036969, A351105.

Sequence in context: A391711 A230692 A267992 * A198183 A259493 A253459

Adjacent sequences: A060490 A060491 A060492 * A060494 A060495 A060496

Keyword

nonn,easy

Author

Larry Reeves (larryr(AT)acm.org), Mar 20 2001

Extensions

Missing a(0)=0 inserted by Alois P. Heinz, Feb 19 2022

Status

approved