A112495 Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.

3, 25, 130, 546, 2037, 7071, 23436, 75328, 237127, 735813, 2260518, 6896046, 20933673, 63325051, 191088976, 575625900, 1731858075, 5206059585, 15640198410, 46966732090, 140996664733, 423191320215, 1269993390420

Offset: 0

Author

Wolfdieter Lang, Oct 14 2005

Keywords

• nonn
• easy

Comments

2*a(n-4) is the number of ternary words of length n where two of the letters are used at least twice. For example, for n=5 the 50 words that use 0 and 1 at least twice are 00011 (10 of this type), 00111 (10 of this type) and 00112 (30 of this type). - Enrique Navarrete, Feb 14 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

Crossrefs

Cf. A000295 (second column).

Column k=2 of A124324 (shifted).

Programs

Mathematica

CoefficientList[Series[(3 - 5*x)/(((1 - x)^3)*((1 - 2*x)^2)*(1 - 3*x)), {x, 0, 50}], x] (* G. C. Greubel, Nov 13 2017 *)
Table[3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2, {n,0,25}] (* Vaclav Kotesovec, Jul 23 2021 *)

PARI

x='x+O('x^50); Vec((3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x))) \\ G. C. Greubel, Nov 13 2017

Formula

a(n) = 3*a(n-1)+ (n+3)*(2^(n+2)-(n+3)), n>=1, a(0)=3.

G.f.: (3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x)).

a(n) = 3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2. - Vaclav Kotesovec, Jul 23 2021

E.g.f.: (1/2)*exp(x)*(exp(x)-x-1)^2 (with offset 4). - Enrique Navarrete, Feb 14 2025