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 (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
FORMULA
G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^12.
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.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^7.
EXAMPLE
First differences: 1, 127, 2059, 14197, 61741, ... (A022523)
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The seventh powers: 1, 128, 2187, 16384, 78125, ... (A001015)
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First partial sums: 1, 129, 2316, 18700, 96825, ... (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ... (A250212)
Third partial sums: 1, 131, 2577, 23723, 141694, ... (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ... (this sequence)
MAPLE
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
MATHEMATICA
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 *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^7]]]] (* or *)
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]
PROG
(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
(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
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Feb 05 2015
STATUS
approved