A062027 - OEIS

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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

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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 *)