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 A254681 Fifth partial sums of fourth powers (A000583). 10
 1, 21, 176, 936, 3750, 12342, 35112, 89232, 207207, 446875, 906048, 1743248, 3206268, 5670588, 9690000, 16062144, 25912029, 40797009, 62837104, 94875000, 140670530, 205134930, 294610680, 417203280, 583171875, 805386231 (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 Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1). FORMULA G.f.:(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^10. a(n) = n^2*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(5 + 2*n)/30240. a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^4. E.g.f.: (1/30240)*exp(x)*(30240 + 604800*x + 2041200*x^2 + 2368800*x^3 + 1233540*x^4 + 326592*x^5 + 46410*x^6 + 3540*x^7 + 135*x^8 + 2*x^9). - Stefano Spezia, Dec 02 2018 EXAMPLE Fourth differences:  1, 12,  23,  24, (repeat 24)  ...   (A101104) Third differences:   1, 13,  36,  60,   84,   108, ...   (A101103) Second differences:  1, 14,  50, 110,  194,   302, ...   (A005914) First differences:   1, 15,  65, 175,  369,   671, ...   (A005917) ------------------------------------------------------------------------- The fourth powers:   1, 16,  81, 256,  625,  1296, ...   (A000583) ------------------------------------------------------------------------- First partial sums:  1, 17,  98, 354,  979,  2275, ...   (A000538) Second partial sums: 1, 18, 116, 470, 1449,  3724, ...   (A101089) Third partial sums:  1, 19, 135, 605, 2054,  5778, ...   (A101090) Fourth partial sums: 1, 20, 155, 760, 2814,  8592, ...   (A101091) Fifth partial sums:  1, 21, 176, 936, 3750, 12342, ...   (this sequence) MAPLE seq(coeff(series((x+11*x^2+11*x^3+x^4)/(1-x)^10, x, n+1), x, n), n = 1 .. 30); # Muniru A Asiru, Dec 02 2018 MATHEMATICA Table[n^2(1+n)(2+n)(3+n)(4+n)(5+n)^2(5+2n)/30240, {n, 26}] (* or *) CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(1-x)^10, {x, 0, 25}], x] CoefficientList[Series[(1/30240)E^x (30240 + 604800 x + 2041200 x^2 + 2368800 x^3 + 1233540 x^4 + 326592 x^5 + 46410 x^6 + 3540 x^7 + 135 x^8 + 2 x^9), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 02 2018 *) PROG (PARI) my(x='x+O('x^30)); Vec((x+11*x^2+11*x^3+x^4)/(1-x)^10) \\ G. C. Greubel, Dec 01 2018 (MAGMA) [Binomial(n+5, 6)*n*(n+5)*(2*n+5)/42: n in [1..30]]; // G. C. Greubel, Dec 01 2018 (Sage) [binomial(n+5, 6)*n*(n+5)*(2*n+5)/42 for n in (1..30)] # G. C. Greubel, Dec 01 2018 CROSSREFS Cf. A000538, A000583, A005914, A005917, A101089, A101090, A101091, A101103, A101104, A254682, A254683, A254684. Sequence in context: A015880 A113163 A090021 * A219625 A244875 A025604 Adjacent sequences:  A254678 A254679 A254680 * A254682 A254683 A254684 KEYWORD nonn,easy AUTHOR Luciano Ancora, Feb 12 2015 STATUS approved

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Last modified August 22 08:27 EDT 2019. Contains 326172 sequences. (Running on oeis4.)