A013069 Expansion of e.g.f.: exp(arcsinh(x)+log(x+1))=1+2*x+3/2!*x^2+3/3!*x^3-3/4!*x^4-15/5!*x^5...

1, 2, 3, 3, -3, -15, 45, 315, -1575, -14175, 99225, 1091475, -9823275, -127702575, 1404728325, 21070924875, -273922023375, -4656674397375, 69850115960625, 1327152203251875, -22561587455281875

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..449

Formula

D-finite with recurrence: n*(n - 2)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n) + (n + 4)*(n + 3)*(n + 2)*(2*n^2 + 2*n - 3)*a(n + 1) + (2*n + 3)*(n + 4)*(n + 3)*(n + 1)*a(n + 2) + 2*(n + 4)*(n + 3)*(n + 1)*a(n + 3) + (n + 4)*(n + 3)*a(n + 4) = 0. - Robert Israel, Feb 23 2026

Maple

f:= gfun:-rectoproc({n*(n - 2)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n) + (n + 4)*(n + 3)*(n + 2)*(2*n^2 + 2*n - 3)*a(n + 1) + (2*n + 3)*(n + 4)*(n + 3)*(n + 1)*a(n + 2) + 2*(n + 4)*(n + 3)*(n + 1)*a(n + 3) + (n + 4)*(n + 3)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 23 2026

Mathematica

With[{nn=20}, CoefficientList[Series[Exp[ArcSinh[x]+Log[x+1]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Aug 15 2024 *)

Crossrefs

Cf. A046126.

a(2n) = (-1)^(n+1) * A079484(n), n>1.

Sequence in context: A347208 A210483 A022296 * A096393 A010120 A012887

Adjacent sequences: A013066 A013067 A013068 * A013070 A013071 A013072

Keyword

sign

Author

Patrick Demichel

Extensions

Definition clarified by Harvey P. Dale, Aug 15 2024

Status

approved