A136659 Unsigned third column (k=2) of triangle A136656 divided by 4.

1, 9, 75, 660, 6300, 65520, 740880, 9072000, 119750400, 1696464000, 25686460800, 414096883200, 7083236160000, 128152088064000, 2445351068160000, 49084865077248000, 1033983353475072000, 22808456326656000000, 525810946517176320000, 12645008187498086400000

Offset: 0

Author

Wolfdieter Lang, Feb 22 2008

Keywords

• nonn
• easy
• changed

Comments

Also unsigned second column of triangle A136657 divided by 2.

References

• Charalambos A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, Table 8.3, p. 311, with s=-2, k=2 column/4.

Crossrefs

Cf. A001710 (1/2 of unsigned k=1 column of A136657). A136660 (k=3 column divided by 8), A136656.

Cf. A001113, A001620, A091725, A099285.

Programs

Mathematica

a[n_] := (n + 8)*(n + 1)*(n + 3)!/48; Array[a, 20, 0] (* Amiram Eldar, Aug 31 2025 *)

Formula

a(n) = |A136656(n+2,2)|/4, n>=0.

E.g.f.: (2+6*x-3*x^2)/(2*(1-x)^6) (derived from the one given for the column k=2 under A136656).

a(n) = (n+4)!/2 * sum((k+1)!/(k+4)!,k=1..n), with offset 1. - Gary Detlefs, Jul 27 2010

a(n) = (1/48) * (n+8)*(n+1)*(n+3)!. - Gary Detlefs, Aug 03 2010

From Amiram Eldar, Aug 31 2025: (Start)

Sum_{n>=0} 1/a(n) = 44836/245 - 480*e/7 - 24*gamma/7 + 24*ExpIntegralEi(1)/7, where e = A001113, gamma = A001620, and ExpIntegralEi(1) = A091725.

Sum_{n>=0} (-1)^n/a(n) = 39724/245 - 3120/(7*e) + 24*gamma/7 - 24*ExpIntegralEi(-1)/7, where ExpIntegralEi(-1) = -A099285. (End)