This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A254640 #38 May 04 2023 12:35:08
%S A254640 1,67,927,6677,32942,126378,404634,1129854,2833479,6515509,13947505,
%T A254640 28115451,53846156,98669156,173975076,296541132,490504893,789878583,
%U A254640 1241708083,1909993393,2880500634,4266609710,6216356510,8920844010,12624212835,17635378761
%N A254640 Third partial sums of sixth powers (A001014).
%C A254640 This is one of a sequence of arrays that are the convolutions of the zero-padded sequences binomial(2n-1+k,k) with the Eulerian polynomials E(n,x) of A008292, represented by E(n,x) (1-x)^(-2n), which generate increasing partial sums of powers of integers:
%C A254640 n= 2) (1 + 4*x + x^2)/(1-x)^4 is the o.g.f. of A000578, the convolution of (1,4,1) with A000292, giving the powers of m^3.
%C A254640 n= 3) (1 + 11*x + 11*x^2 + x^3)/(1-x)^6 is the o.g.f. of A000538, convolution of (1,11,11,1) with A000389, giving the partial sums of m^4.
%C A254640 n= 4) (1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^8, the o.g.f. of A101092, convolution of (1,26,66,26,1) with A000580, the second partial sums of m^5
%C A254640 n= 5) (1 + 57*x + 302*x^2 + 302*x^3 + 57*x^4 + x^5)/(1-x)^10, the o.g.f. of A254460, convolution of (1,57,302,302,57,1) with A000582, giving the third partial sums of m^6. - _Tom Copeland_, Dec 07 2015
%H A254640 Colin Barker, <a href="/A254640/b254640.txt">Table of n, a(n) for n = 1..1000</a>
%H A254640 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>
%H A254640 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A254640 a(n) = n*(1+n)*(2+n)*(3+n)*(3+2*n)*(2 -30*n +35*n^2 +30*n^3 +5*n^4)/5040.
%F A254640 G.f.: x*(1+x)*(1 +56*x +246*x^2 +56*x^3 +x^4)/(1-x)^10. - _Colin Barker_, Feb 04 2015
%p A254640 seq(binomial(n+3,4)*(2*n+3)*(5*n^4 +30*n^3 +35*n^2 -30*n +2)/210, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t A254640 Table[n(1+n)(2+n)(3+n)(3+2n)(2 -30n +35n^2 +30n^3 +5n^4)/5040, {n, 30}] (* or *) CoefficientList[Series[(x+1)(x^4 +56x^3 +246x^2 +56x +1)/(x - 1)^10, {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 05 2015 *)
%o A254640 (PARI) vector(30, n, n*(1+n)*(2+n)*(3+n)*(3+2*n)*(2-30*n+35*n^2+30*n^3+5*n^4)/5040) \\ _Colin Barker_, Feb 04 2015
%o A254640 (Magma) [n*(1+n)*(2+n)*(3+n)*(3+2*n)*(2-30*n+35*n^2+30*n^3+ 5*n^4)/5040: n in [1..30]]; // _Vincenzo Librandi_, Feb 05 2015
%o A254640 (Sage) [binomial(n+3,4)*(2*n+3)*(5*n^4 +30*n^3 +35*n^2 -30*n +2)/210 for n in (1..30)] # _G. C. Greubel_, Aug 28 2019
%o A254640 (GAP) List([1..30], n-> Binomial(n+3,4)*(2*n+3)*(5*n^4 +30*n^3 +35*n^2 -30*n +2)/210); # _G. C. Greubel_, Aug 28 2019
%o A254640 (Python)
%o A254640 def A254640(n): return n*(n*(n*(n*(n*(n*(n*(n*(10*n + 135) + 720) + 1890) + 2394) + 945) - 640) - 450) + 36)//5040 # _Chai Wah Wu_, Dec 07 2021
%Y A254640 Cf. A008292, A000578, A000292, A000538, A000389, A101092, A000580, A254460, A000582.
%K A254640 nonn,easy
%O A254640 1,2
%A A254640 _Luciano Ancora_, Feb 04 2015