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 A253637 Second partial sums of ninth powers (A001017). 4
 1, 514, 20710, 303050, 2538515, 14851676, 67518444, 254402940, 828707925, 2403012910, 6335265586, 15427298614, 35123831015, 75481410200, 154282348760, 301802764056, 567911055849, 1032378638010, 1819533917950, 3118689197890, 5212124524411, 8512829068724, 13614686274500, 21358351020500, 32916713032125, 49904578722726 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas. LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Luciano Ancora, Recurrence relation for the second partial sums of m-th powers Luciano Ancora, Second partial sums of the m-th powers Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1). FORMULA a(n) = n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4 + 24*n^3 + 5*n^2 - 38*n + 25)/660. a(n) = 2*a(n-1) - a(n-2) + n^9. G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1-x)^12. - Vincenzo Librandi, Jan 19 2015 MAPLE seq(n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3+5*n^2-38*n+ 25)/660, n=1..30); # G. C. Greubel, Aug 28 2019 MATHEMATICA CoefficientList[Series[(1 +502x +14608x^2 +88234x^3 +156190x^4 +88234x^5 +14608x^6 +502x^7 +x^8)/(1-x)^12, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 19 2015 *) Nest[Accumulate, Range[30]^9, 2] (* Harvey P. Dale, Apr 18 2021 *) PROG (PARI) a(n) = (6*n^11 + 66*n^10 + 275*n^9 + 495*n^8 + 198*n^7 - 462*n^6 - 330*n^5 + 330*n^4 + 231*n^3 - 99*n^2 - 50*n)/660; \\ Michel Marcus, Jan 08 2015 (Magma) [n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3+5*n^2-38*n+ 25)/660: n in [1..30]]; // G. C. Greubel, Aug 28 2019 (Sage) [n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3+5*n^2-38*n+ 25)/660 for n in (1..30)] # G. C. Greubel, Aug 28 2019 (GAP) List([1..30], n-> n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3 +5*n^2-38*n+ 25)/660 ); # G. C. Greubel, Aug 28 2019 CROSSREFS Cf. A001017. Sequence in context: A045054 A301547 A229331 * A231415 A271760 A228999 Adjacent sequences: A253634 A253635 A253636 * A253638 A253639 A253640 KEYWORD nonn,easy AUTHOR Luciano Ancora, Jan 07 2015 STATUS approved

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Last modified July 13 12:36 EDT 2024. Contains 374284 sequences. (Running on oeis4.)