A135038 - OEIS

%I #21 Oct 21 2022 21:41:39

%S 0,840,8760,41520,134544,347064,768264,1523424,2780064,4754088,

%T 7715928,11996688,17994288,26179608,37102632,51398592,69794112,

%U 93113352,122284152,158344176,202447056,255868536,320012616,396417696,486762720

%N Sums of the products of n consecutive quadruples of numbers.

%H G. C. Greubel, <a href="/A135038/b135038.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, -15, 20, -15, 6, -1).

%F a(n) = floor(51.2*n^5 - 32*n^4 - 48*n^3 + 20*n^2 + 8.8*n).

%F 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).

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

%F 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

%F From _G. C. Greubel_, Sep 17 2016: (Start)

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

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

%e 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.

%p 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

%t 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 *)

%t 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 *)

%o (PARI) sumprod3(n)= local(x,s=0); forstep(x=0,n,4, s+=x*(x+1)*(x+2)*(x+3); print1(s",") )

%o (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

%K nonn,easy

%O 1,2

%A _Cino Hilliard_, Feb 11 2008