A250212 Second partial sums of seventh powers (A001015).

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

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

Luciano Ancora, Recurrence relation for the second partial sums of m-th powers

Luciano Ancora, Second partial sums of the m-th powers

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

Adjacent sequences: A250209 A250210 A250211 * A250213 A250214 A250215

Keyword

nonn,easy

Author

Luciano Ancora, Jan 18 2015

Status

approved