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 A254646 #53 Sep 08 2022 08:46:11
%S A254646 1,132,2709,26432,168126,804552,3136014,10459968,30856839,82407052,
%T A254646 202678203,465069696,1005729452,2066218896,4058958828,7664805504,
%U A254646 13974953853,24692818836,42415687153,71020845504,116186669130,186085891160,292296070170,450981236160,684408934755
%N A254646 Fourth partial sums of seventh powers (A001015).
%H A254646 Luciano Ancora, <a href="/A254646/b254646.txt">Table of n, a(n) for n = 1..1000</a>
%H A254646 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>
%H A254646 Luciano Ancora, <a href="/A254646/a254646_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>
%H A254646 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F A254646 G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^12.
%F A254646 a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(48 - 100*n - 89*n^2 + 160*n^3 + 140*n^4 + 36*n^5 + 3*n^6)/23760.
%F A254646 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^7.
%e A254646 First differences: 1, 127, 2059, 14197, 61741, ... (A022523)
%e A254646 ----------------------------------------------------------------------
%e A254646 The seventh powers: 1, 128, 2187, 16384, 78125, ... (A001015)
%e A254646 ----------------------------------------------------------------------
%e A254646 First partial sums: 1, 129, 2316, 18700, 96825, ... (A000541)
%e A254646 Second partial sums: 1, 130, 2446, 21146, 117971, ... (A250212)
%e A254646 Third partial sums: 1, 131, 2577, 23723, 141694, ... (A254641)
%e A254646 Fourth partial sums: 1, 132, 2709, 26432, 168126, ... (this sequence)
%p A254646 seq(binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t A254646 Table[n (1 + n) (2 + n) (3 + n) (4 + n) (48 - 100 n - 89 n^2 + 160 n^3 + 140 n^4 + 36 n^5 + 3 n^6)/23760, {n, 20}] (* or *)
%t A254646 Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^7]]]] (* or *)
%t A254646 CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(- 1 + x)^12, {x, 0, 19}], x]
%o A254646 (PARI) a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(48-100*n-89*n^2+160*n^3+140*n^4 +36*n^5+3*n^6)/23760 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A254646 (Magma) [Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198: n in [1..30]]; // _G. C. Greubel_, Aug 28 2019
%o A254646 (Sage) [binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198 for n in (1..30)] # _G. C. Greubel_, Aug 28 2019
%o A254646 (GAP) List([1..30], n-> Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198); # _G. C. Greubel_, Aug 28 2019
%Y A254646 Cf. A000541, A001015, A022523, A250212, A254641, A254645.
%K A254646 nonn,easy
%O A254646 1,2
%A A254646 _Luciano Ancora_, Feb 05 2015