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 A135038 Sums of the products of n consecutive quadruples of numbers. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 AUTHOR Cino Hilliard, Feb 11 2008 STATUS approved

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Last modified September 20 04:04 EDT 2020. Contains 337264 sequences. (Running on oeis4.)