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
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
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
Crossrefs
Cf. A239094 (same sequence, shifted by 1).
Programs
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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
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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
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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
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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 *)
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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
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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
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
Comments