A062027 a(1) = a(2) = a(3) = 1 and a(n) = 24*binomial(n+1, 5) + n*(n^2 - n + 6) for n > 3.

1, 1, 1, 96, 274, 720, 1680, 3520, 6750, 12048, 20284, 32544, 50154, 74704, 108072, 152448, 210358, 284688, 378708, 496096, 640962, 817872, 1031872, 1288512, 1593870, 1954576, 2377836, 2871456, 3443866, 4104144, 4862040, 5728000

Offset

1,4

Comments

Previous name: a(1) = a(2) = a(3) = 1; and for n>3 a(n) = 1*2*3*4 + 2*3*4*5 + 3*4*5*6 + ... + (n-1)*n*1*2 + n*1*2*3, the sum of the cyclic product of terms taken four at a time, final term being n*1*2*3 = 6n.

Links

Harry J. Smith, Table of n, a(n) for n=1..1000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = (n+1)*(n)*(n-1)*(n-2)*(n-3)/5 + n*(n^2 - n + 6), for n>3.

From G. C. Greubel, May 05 2022: (Start)

G.f.: -5*x*(1 + 3*x + 7*x^2) + 2*x*(3 - 10*x + 15*x^2 + 4*x^4)/(1-x)^6.

E.g.f.: (1/5)*x*(30 + 10*x + 5*x^2 + 5*x^3 + x^4)*exp(x) - (5/6)*x*(6 + 9*x + 7*x^2). (End)

Example

a(5) = 1*2*3*4 + 2*3*4*5 + 3*4*5*1 + 4*5*1*2 + 5*1*2*3 = 274.

Mathematica

Table[24*Binomial[n+1, 5] +n*(n^2-n+6) -5*(2^n-1)*Boole[n<4], {n, 40}] (* G. C. Greubel, May 05 2022 *)

Prog

(PARI) { a=1; for(n=1, 100, if (n>3, a=(n+1)*n*(n-1)*(n-2)*(n-3)/5 +n*(n^2-n+6)); write("b062027.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 30 2009
(SageMath) [24*binomial(n+1, 5) +n*(n^2-n+6) -5*(2^n-1)*bool(n<4) for n in (1..40)] # G. C. Greubel, May 05 2022

Crossrefs

Sequence in context: A232939 A202591 A202584 * A320883 A048189 A304830

Adjacent sequences: A062024 A062025 A062026 * A062028 A062029 A062030

Keyword

nonn,easy

Author

Amarnath Murthy, Jun 02 2001

Extensions

More terms from Jason Earls, Jun 07 2001

Term a(4) corrected by Harry J. Smith, Jul 30 2009

Name changed by G. C. Greubel, May 05 2022

Status

approved