A254643 Third partial sums of ninth powers (A001017).

1, 515, 21225, 324275, 2862790, 17714466, 85232910, 339635850, 1168343775, 3571356685, 9906622271, 25333920885, 60457751900, 135939162100, 290221510860, 592024274916, 1159935330765, 2192313968775, 4011847886725, 7130537084615

Offset: 1

Author

Luciano Ancora, Feb 05 2015

Keywords

• nonn
• easy

Examples

			First differences:   1, 511, 19171, 242461, 1690981, ... (A022525)
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The ninth powers:    1, 512, 19683, 262144, 1953125, ... (A001017)
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First partial sums:  1, 513, 20196, 282340, 2235465, ... (A007487)
Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637)
Third partial sums:  1, 515, 21225, 324275, 2862790, ... (this sequence)
		

Links

• Luciano Ancora, Table of n, a(n) for n = 1..1000
• Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials
• Luciano Ancora, Pascal’s triangle and recurrence relations for partial sums of m-th powers
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A001017, A007487, A022525, A253637.

Programs

GAP

List([1..30], n-> Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110); # G. C. Greubel, Aug 28 2019

Magma

[Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110: n in [1..30]]; // G. C. Greubel, Aug 28 2019

Maple

seq(binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110, n=1..30); # G. C. Greubel, Aug 28 2019

Mathematica

Table[n(1+n)(2+n)(3+n)(-50 +84n +127n^2 -204n^3 -97n^4 +126n^5 +98n^6 +24n^7 +2n^8)/2640, {n, 20}] (* or *)
CoefficientList[Series[(1 +502x +14608x^2 +88234x^3 +156190x^4 +88234x^5 +14608x^6 +502x^7 +x^8)/(1-x)^13, {x, 0, 19}], x] (* Ancora *)
Accumulate[Accumulate[Accumulate[Range[10]^9]]] (* Alonso del Arte, Feb 09 2015 *)

PARI

vector(30, n, m=n+3; binomial(m,4)*(2*(n*m)^4 -10*(n*m)^3 +11*(n*m)^2 +28*(n*m) -50)/110) \\ G. C. Greubel, Aug 28 2019

Sage

[binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: x*(1 +502*x +14608*x^2 +88234*x^3 +156190*x^4 +88234*x^5 +14608*x^6 +502*x^7 +x^8)/(1-x)^13.

a(n) = n*(1+n)*(2+n)*(3+n)*(-50 +84*n +127*n^2 -204*n^3 -97*n^4 +126*n^5 +98*n^6 +24*n^7 +2*n^8)/2640.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^9.

Extensions

Edited by Alonso del Arte and Bruno Berselli, Feb 10 2015