A254872 Seventh partial sums of sixth powers (A001014).
1, 71, 1205, 11075, 70295, 345857, 1409387, 4962365, 15539750, 44192010, 115917118, 283828498, 654885730, 1434717550, 3002927770, 6035661334, 11699568079, 21951176425, 39988722875, 70920437325, 122735050305
Offset: 1
Examples
First differences: 1, 63, 665, 3367, 11529, ... (A022522) -------------------------------------------------------------------- The sixth powers: 1, 64, 729, 4096, 15625, ... (A001014) -------------------------------------------------------------------- First partial sums: 1, 65, 794, 4890, 20515, ... (A000540) Second partial sums: 1, 66, 860, 5750, 26265, ... (A101093) Third partial sums: 1, 67, 927, 6677, 32942, ... (A254640) Fourth partial sums: 1, 68, 995, 7672, 40614, ... (A254645) Fifth partial sums: 1, 69, 1064, 8736, 49350, ... (A254683) Sixth partial sums: 1, 70, 1134, 9870, 59220, ... (A254472) Seventh partial sums: 1, 71, 1205, 11075, 70295, ... (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 (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).
Crossrefs
Programs
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Mathematica
Table[(n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n) (- 49 + 147 n^2 + 42 n^3 + 3 n^4))/51891840, {n, 21}] (* or *) CoefficientList[Series[(1 + 57 x + 302 x^2 + 302 x^3 + 57 x^4 + x^5)/(- 1 + x)^14, {x, 0, 20}], x]
Formula
G.f.: (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)/(- 1 + x)^14.
a(n) = (n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(- 49 + 147*n^2 + 42*n^3 + 3*n^4))/51891840.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^6.