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 A254644 Fourth partial sums of fifth powers (A000584). 8
 1, 36, 381, 2336, 10326, 36552, 110022, 292512, 704847, 1567852, 3263403, 6422208, 12046268, 21675408, 37608828, 63194304, 103199469, 164281524, 255573769, 389409504, 582206130, 855534680, 1237402530, 1763779680, 2480401755, 3444885756, 4729197591, 6422513536, 8634521016, 11499207456 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Luciano Ancora, Table of n, a(n) for n = 1..1000 Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials Luciano Ancora, Pascal’s triangle and recurrence relations  for partial sums of m-th powers Index entries for linear recurrences with constant coefficients, signature(10,-45,120,-210,252,-210,120,-45,10,-1). FORMULA G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^10. a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(-24 + 20*n + 85*n^2 + 40*n^3 + 5*n^4)/15120. a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^5. EXAMPLE Fifth differences: 1, 27, 93, 119, 120, (repeat 120) (A101100) Fourth differences: 1, 28, 121, 240, 360, 480, ... (A101095) Third differences: 1, 29, 150, 390, 750, 1230, ... (A101096) Second differences: 1, 30, 180, 570, 1320, 2550, ... (A101098) First differences: 1, 31, 211, 781, 2101, 4651, ... (A022521) ------------------------------------------------------------------------- The fifth powers: 1, 32, 243, 1024, 3125, 7776, ... (A000584) ------------------------------------------------------------------------- First partial sums: 1, 33, 276, 1300, 4425, 12201, ... (A000539) Second partial sums: 1, 34, 310, 1610, 6035, 18236, ... (A101092) Third partial sums: 1, 35, 345, 1955, 7990, 26226, ... (A101099) Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (this sequence) MAPLE seq(binomial(n+4, 5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126, n=1..30); # G. C. Greubel, Aug 28 2019 MATHEMATICA Table[n(1+n)(2+n)(3+n)(4+n)(-24 +20n +85n^2 +40n^3 +5n^4)/15120, {n, 30}] (* or *) Accumulate[Accumulate[Accumulate[Accumulate[Range[24]^5]]]] (* or *) CoefficientList[Series[(1 +26x +66x^2 +26x^3 +x^4)/(1-x)^10, {x, 0, 30}], x] Nest[Accumulate, Range[30]^5, 4] (* or *) LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 36, 381, 2336, 10326, 36552, 110022, 292512, 704847, 1567852}, 30] (* Harvey P. Dale, May 08 2016 *) PROG (PARI) vector(30, n, m=n+2; binomial(m+2, 5)*(5*m^4 -35*m^2 +36)/126) \\ G. C. Greubel, Aug 28 2019 (Magma) [Binomial(n+4, 5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126: n in [1..30]]; // G. C. Greubel, Aug 28 2019 (Sage) [binomial(n+4, 5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126 for n in (1..30)] # G. C. Greubel, Aug 28 2019 (GAP) List([1..30], n-> Binomial(n+4, 5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126); # G. C. Greubel, Aug 28 2019 CROSSREFS Cf. A000539, A000584, A022521, A101092, A101095, A101096, A101098, A101099, A101100. Cf. A101091 (fourth partial sums of fourth powers). Sequence in context: A071232 A135828 A250805 * A034686 A160280 A307959 Adjacent sequences: A254641 A254642 A254643 * A254645 A254646 A254647 KEYWORD nonn,easy AUTHOR Luciano Ancora, Feb 05 2015 EXTENSIONS Edited by Bruno Berselli, Feb 10 2015 STATUS approved

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Last modified June 10 19:34 EDT 2023. Contains 363207 sequences. (Running on oeis4.)
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