A030280 - OEIS

%I #28 Jun 30 2026 22:29:54

%S 1,6,30,137,588,2415,9600,37209,141336,528049,1945800,7086555,

%T 25550547,91319973,323890062,1140985116,3995155506,13913280006,

%U 48216776266,166355744244,571636515369,1957015021744,6677155551312,22710546483471,77020249727449,260504237234973,878898766586658

%N COMPOSE triangular numbers with triangular numbers.

%H Harvey P. Dale, <a href="/A030280/b030280.txt">Table of n, a(n) for n = 1..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (12, -57, 139, -195, 174, -102, 39, -9, 1).

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

%F 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

%F From _Seiichi Manyama_, May 19 2026: (Start)

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

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

%t CoefficientList[Series[((1-x)^6)/((1-4x+3x^2-x^3)^3),{x,0,30}],x] (* _Harvey P. Dale_, Jan 08 2013 *)

%t (* Alternative: *)

%t 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 *)

%Y Cf. A030267, A052544, A279283, A396209.

%K nonn,nice,easy,changed

%O 1,2

%A _Christian G. Bower_