login
Second partial sums of seventh powers (A001015).
5

%I #110 Sep 08 2022 08:46:10

%S 1,130,2446,21146,117971,494732,1695036,4992492,13072917,31153342,

%T 68720938,142120342,278268263,519829688,932250488,1613106744,

%U 2704301673,4407716634,7005003334,10882290034,16560665275,24733398404,36310956980,52474986980,74742532605,105041888406

%N Second partial sums of seventh powers (A001015).

%C 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.

%H G. C. Greubel, <a href="/A250212/b250212.txt">Table of n, a(n) for n = 1..1000</a>

%H Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>

%H Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = n*(n+1)*(n+2)*(5*n^6 + 30*n^5 + 50*n^4 - 37*n^2 + 6*n + 6)/360.

%F a(n) = 2*a(n-1) - a(n-2) + n^7.

%F 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

%F a(n) = A239094(n+1). - _Danny Rorabaugh_, Apr 22 2015

%p 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

%t Accumulate[Accumulate[Range[25]^7]] (* _Robert G. Wilson v_, Jan 21 2015 *)

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

%t 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 *)

%t 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 *)

%o (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

%o (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

%o (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

%o (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

%Y Cf. A239094 (same sequence, shifted by 1).

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Jan 18 2015