A016223 Expansion of 1/((1-x) * (1-4*x) * (1-7*x)).

1, 12, 105, 820, 6081, 43932, 312985, 2212740, 15576561, 109385452, 767096265, 5375266260, 37649233441, 263634112572, 1845796701945, 12922008569380, 90459786608721, 633241412753292, 4432781515242025, 31029837110570100, 217210325789494401, 1520478144588475612

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (12,-39,28).

Formula

a(n) = (1/18) - (16/9)*4^n + (49/18)*7^n. - Antonio Alberto Olivares, Feb 07 2010 [corrected by Seiichi Manyama, May 03 2025]

a(0)=1, a(1)=12, a(n) = 11*a(n-1) - 28*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011

E.g.f.: exp(x)*(1 - 32*exp(3*x) + 49*exp(6*x))/(2!*3^2). - This is (d^2/dx^2) (exp(x)*(exp(x) - 1)^2 / (2*3^2)). See also the second column of the Sheffer triangle A282629 divided by 3^2. - Wolfdieter Lang, Apr 08 2017

From Seiichi Manyama, May 03 2025: (Start)

a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k+2) * Stirling2(k+2,2).

G.f.: B(x)^3, where B(x) is the g.f. of A383627. (End)

Maple

a:=n->sum((7^(n+1-j)-4^(n+1-j))/3, j=0..n+1): seq(a(n), n=0..20); # Zerinvary Lajos, Jan 15 2007

Mathematica

LinearRecurrence[{12, -39, 28}, {1, 12, 105}, 25] (* Paolo Xausa, May 10 2026 *)
CoefficientList[Series[1/((1-x)(1-4x)(1-7x)), {x, 0, 30}], x] (* Harvey P. Dale, May 12 2026 *)

Prog

(PARI) a(n) = (1-2*4^(n+2)+7^(n+2))/18; \\ Seiichi Manyama, May 03 2025

Crossrefs

Cf. A002450, A021874, A111577.

Cf. A282629, A383627.

Sequence in context: A157024 A051815 A004321 * A027142 A090816 A384668

Adjacent sequences: A016220 A016221 A016222 * A016224 A016225 A016226

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved