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A250212 Second partial sums of seventh powers (A001015). 5
1, 130, 2446, 21146, 117971, 494732, 1695036, 4992492, 13072917, 31153342, 68720938, 142120342, 278268263, 519829688, 932250488, 1613106744, 2704301673, 4407716634, 7005003334, 10882290034, 16560665275, 24733398404, 36310956980, 52474986980, 74742532605, 105041888406 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The general formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) is the m-th Faulhaber’s polynomial.
LINKS
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
a(n) = n*(n+1)*(n+2)*(5*n^6 + 30*n^5 + 50*n^4 - 37*n^2 + 6*n + 6)/360.
a(n) = 2*a(n-1) - a(n-2) + n^7.
G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^10. - Georg Fischer, May 24 2019
a(n) = A239094(n+1). - Danny Rorabaugh, Apr 22 2015
MAPLE
seq(binomial(n+2, 3)*(5*(n+1)^6 -25*(n+1)^4 +38*(n+1)^2 -12)/60, n=1..30); # G. C. Greubel, Aug 28 2019
MATHEMATICA
Accumulate[Accumulate[Range[25]^7]] (* Robert G. Wilson v, Jan 21 2015 *)
Table[(n(n+1)(n+2)(5n^6+30n^5+50n^4-37n^2+6n+6)/360), {n, 30}] (* Vincenzo Librandi, Jan 22 2015 *)
RecurrenceTable[{a[n]==2a[n-1]-a[n-2]+n^7, a[1]==1, a[2]==130}, a, {n, 30}] (* Bruno Berselli, Jan 22 2015 *)
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 130, 2446, 21146, 117971, 494732, 1695036, 4992492, 13072917, 31153342}, 30] (* Harvey P. Dale, Jan 19 2020 *)
PROG
(PARI) vector(50, n, n*(n+1)*(n+2)*(5*n^6 + 30*n^5 + 50*n^4 - 37*n^2 + 6*n + 6)/360) \\ Michel Marcus, Jan 21 2015
(Magma) [(n*(n + 1)*(n + 2)*(5*n^6 + 30*n^5 + 50*n^4 -37*n^2 + 6*n + 6) / 360): n in [1..30]]; // Vincenzo Librandi, Jan 22 2015
(Sage) [binomial(n+2, 3)*(5*(n+1)^6 -25*(n+1)^4 +38*(n+1)^2 -12)/60 for n in (1..30)] # G. C. Greubel, Aug 28 2019
(GAP) List([1..30], n-> Binomial(n+2, 3)*(5*(n+1)^6 -25*(n+1)^4 +38*(n+ 1)^2 -12)/60); # G. C. Greubel, Aug 28 2019
CROSSREFS
Cf. A239094 (same sequence, shifted by 1).
Sequence in context: A301545 A229329 A262108 * A239094 A084641 A271758
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Jan 18 2015
STATUS
approved

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Last modified June 18 12:25 EDT 2024. Contains 373481 sequences. (Running on oeis4.)