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A250212 Second partial sums of seventh powers (A001015). 4
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

Table of n, a(n) for n=1..26.

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

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

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 + 1171*x^2 + 2616*x^3 + 291*x^4 + 2520*x^5 - 4199*x^6 + 5040*x^7 - 4200*x^8 + 2400*x^9 -900*x^10 + 200*x^11 - 20*x^12) / (1 - x)^10. - Vincenzo Librandi, Jan 22 2015

a(n) = A239094(n+1). - Danny Rorabaugh, Apr 22 2015

MATHEMATICA

Accumulate[Accumulate[Range[25]^7]] (* Robert G. Wilson v, Jan 21 2015 *)

Table[(n (n+1) (n+2) (5 n^6 + 30 n^5 + 50 n^4 -37 n^2 + 6 n + 6) / 360), {n, 30}] (* Vincenzo Librandi, Jan 22 2015 *)

RecurrenceTable[{a[n] == 2 a[n - 1] - a[n - 2] + n^7, a[1] == 1, a[2] == 130}, a, {n, 1, 30}] (* Bruno Berselli, Jan 22 2015 *)

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

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

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Last modified November 14 06:49 EST 2018. Contains 317162 sequences. (Running on oeis4.)