A030280 COMPOSE triangular numbers with triangular numbers.

1, 6, 30, 137, 588, 2415, 9600, 37209, 141336, 528049, 1945800, 7086555, 25550547, 91319973, 323890062, 1140985116, 3995155506, 13913280006, 48216776266, 166355744244, 571636515369, 1957015021744, 6677155551312, 22710546483471, 77020249727449, 260504237234973, 878898766586658

Offset

1,2

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

N. J. A. Sloane, Transforms

Index entries for linear recurrences with constant coefficients, signature (12, -57, 139, -195, 174, -102, 39, -9, 1).

Formula

G.f.: x * (1-x)^6 / (1-4*x+3*x^2-x^3)^3. [Corrected by Seiichi Manyama, May 19 2026]

a(1)=1, a(2)=6, a(3)=30, a(4)=137, a(5)=588, a(6)=2415, a(7)=9600, a(8)=37209, a(9)=141336, a(n)=12*a(n-1)-57*a(n-2)+139*a(n-3)- 195*a(n-4)+ 174*a(n-5)-102*a(n-6)+39*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Jan 08 2013

From Seiichi Manyama, May 19 2026: (Start)

G.f.: x * B(x)^3, where B(x) is the g.f. of A052544.

a(n) = Sum_{k=1..n} binomial(k+1,2) * binomial(n+2*k-1,3*k-1). (End)

Mathematica

CoefficientList[Series[((1-x)^6)/((1-4x+3x^2-x^3)^3), {x, 0, 30}], x] (* Harvey P. Dale, Jan 08 2013 *)
(* Alternative: *)
LinearRecurrence[{12, -57, 139, -195, 174, -102, 39, -9, 1}, {1, 6, 30, 137, 588, 2415, 9600, 37209, 141336}, 30] (* Harvey P. Dale, Jan 08 2013 *)

Crossrefs

Cf. A030267, A052544, A279283, A396209.

Sequence in context: A162743 A224290 A081895 * A034545 A002920 A255463

Adjacent sequences: A030277 A030278 A030279 * A030281 A030282 A030283

Keyword

nonn,nice,easy,changed

Author

Christian G. Bower

Status

approved