A367591 Expansion of 1/((1-x) * (1-3*x)^3).

1, 10, 64, 334, 1549, 6652, 27064, 105796, 401041, 1483606, 5380840, 19198306, 67559437, 234963352, 808919632, 2760370984, 9346519297, 31429487170, 105039380080, 349114288150, 1154561484781, 3801030845140, 12462203297224, 40705156945324, 132494756301649

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (10,-36,54,-27).

Formula

a(n) = ((2*n^2+4*n+3) * 3^(n+1) - 1)/8.

From Enrique Navarrete, Dec 02 2025: (Start)

a(n) = 10*a(n-1) - 36*a(n-2) + 54*a(n-3) - 27*a(n-4).

E.g.f.: ((54*x^2 + 54*x + 9)*exp(3*x) - exp(x))/8. (End)

Mathematica

CoefficientList[Series[1/((1-x)(1-3x)^3), {x, 0, 30}], x] (* or  *) LinearRecurrence[{10, -36, 54, -27}, {1, 10, 64, 334}, 30] (* Harvey P. Dale, May 16 2026 *)

Prog

(PARI) a(n) = ((2*n^2+4*n+3)*3^(n+1)-1)/8;

Crossrefs

Partial sums of A027472; second partial sums of A383136.

Cf. A055580, A367592.

Sequence in context: A245068 A055855 A061183 * A241401 A219934 A155618

Adjacent sequences: A367588 A367589 A367590 * A367592 A367593 A367594

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 24 2023

Status

approved