A052161 Partial sums of A014825, second partial sums of A002450.

1, 7, 34, 146, 599, 2417, 9696, 38820, 155325, 621355, 2485486, 9942022, 39768179, 159072821, 636291404, 2545165752, 10180663161, 40722652815, 162890611450, 651562446010, 2606249784271, 10424999137337, 41699996549624, 166799986198796, 667199944795509, 2668799779182387, 10675199116729926

Offset

0,2

Comments

From Enrique Navarrete, Nov 08 2025: (Start)

Convolution of the powers of 4 with the triangular numbers [1, 3, 6, 10, ...].

Equivalently, third partial sums of A000302. (End)

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,-15,13,-4).

Formula

a(n) = (2^(2*n+7) - (9*n^2 + 51*n + 74))/54.

a(n) = 4*a(n-1) + C(n+2,2); a(0)=1.

a(n) = Sum_{k=0..n} binomial(n+3, k+3)*3^k. - Paul Barry, Aug 20 2004

G.f.: 1/((1-x)^3*(1-4*x)). - Colin Barker, Jan 12 2012

a(n) = 7*a(n-1) - 15*a(n-2) + 13*a(n-3) - 4*a(n-4), n >= 4. - Enrique Navarrete, Nov 02 2025

E.g.f.: exp(x)*(128*exp(3*x) - 9*x^2 - 60*x - 74)/54. - Enrique Navarrete, Nov 04 2025

Mathematica

CoefficientList[Series[1/((1-x)^3*(1-4*x)), {x, 0, 25}], x] (* Vincenzo Librandi, Apr 28 2012 *)

Prog

(Magma) [((2^(2*n+7))-(9*(n^2)+51*n+74))/54: n in [0..25]]; // Vincenzo Librandi, Apr 28 2012

Crossrefs

Cf. A014825, A002450, A000302, A000217.

Sequence in context: A094891 A306376 A192803 * A080960 A243414 A335087

Adjacent sequences: A052158 A052159 A052160 * A052162 A052163 A052164

Keyword

easy,nonn

Author

Barry E. Williams, Jan 25 2000

Status

approved