A254681 Fifth partial sums of fourth powers (A000583).

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

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 + 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

From Amiram Eldar, Jan 26 2022: (Start)

Sum_{n>=1} 1/a(n) = 172032*log(2)/125 - 2382233/2500.

Sum_{n>=1} (-1)^(n+1)/a(n) = 42*Pi^2/25 - 43008*Pi/125 + 2663213/2500. (End)

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)
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The fourth powers:   1, 16,  81, 256,  625,  1296, ...   (A000583)
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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 *)
Nest[Accumulate[#]&, Range[30]^4, 5] (* Harvey P. Dale, Jan 03 2022 *)

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