A135038 Sums of the products of n consecutive quadruples of numbers.

0, 840, 8760, 41520, 134544, 347064, 768264, 1523424, 2780064, 4754088, 7715928, 11996688, 17994288, 26179608, 37102632, 51398592, 69794112, 93113352, 122284152, 158344176, 202447056, 255868536, 320012616, 396417696, 486762720

Offset

1,2

Links

G. C. Greubel, 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) = floor(51.2*n^5 - 32*n^4 - 48*n^3 + 20*n^2 + 8.8*n).

a(1) = 0*1*2*3, a(2) = 0*1*2*3 + 4*5*6*7, ..., a(n) = 0*1*2*3 + 4*5*6*7 + 8*9*10*11 + ... + (2n-1)*(2n)*(2n+1)(2n+2).

O.g.f.: 24*x^2*(35 +155*x +65*x^2 +x^3)/(1-x)^6 . - R. J. Mathar, Feb 14 2008

a(1)=0, a(2)=840, a(3)=8760, a(4)=41520, a(5)=134544, a(6)=347064, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, May 09 2015

From G. C. Greubel, Sep 17 2016: (Start)

a(n) = (100*n^2 + 44*n - 240*n^3 - 160*n^4 + 256*n^5)/5.

E.g.f.: (4/5)*x^2*(525 + 1300*x + 600*x^2 + 64*x^3)*exp(x). (End)

Example

For n = 3, the sum of the first 3 quadruples is 0*1*2*3 + 4*5*6*7 + 8*9*10*11 = 8760, the 3rd entry in the sequence.

Maple

A135038 := proc(n) 20*n^2+44/5*n-48*n^3-32*n^4+256/5*n^5 ; end: seq(A135038(n), n=1..20) ; # R. J. Mathar, Feb 14 2008

Mathematica

Accumulate[Times@@@Table[4n+Range[0, 3], {n, 0, 30}]] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {0, 840, 8760, 41520, 134544, 347064}, 30] (* Harvey P. Dale, May 09 2015 *)
Table[(100*n^2 + 44*n - 240*n^3 - 160*n^4 + 256*n^5)/5, {n, 1, 25}] (* G. C. Greubel, Sep 17 2016 *)

Prog

(PARI) sumprod3(n)= local(x, s=0); forstep(x=0, n, 4, s+=x*(x+1)*(x+2)*(x+3); print1(s", ") )
(Magma) [(100*n^2+44*n-240*n^3-160*n^4+256*n^5)/5: n in [1..40]]; // Vincenzo Librandi, Sep 18 2016

Crossrefs

Sequence in context: A335141 A177021 A276161 * A215230 A171260 A166758

Adjacent sequences: A135035 A135036 A135037 * A135039 A135040 A135041

Keyword

nonn,easy

Author

Cino Hilliard, Feb 11 2008

Status

approved