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

0,2

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.

Links

Table of n, a(n) for n=0..19.

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)

Mathematica

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

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.

Sequence in context: A254664 A223204 A351513 * A231592 A383163 A335345

Adjacent sequences: A136656 A136657 A136658 * A136660 A136661 A136662

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 22 2008

Status

approved