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A254872 Seventh partial sums of sixth powers (A001014). 5
1, 71, 1205, 11075, 70295, 345857, 1409387, 4962365, 15539750, 44192010, 115917118, 283828498, 654885730, 1434717550, 3002927770, 6035661334, 11699568079, 21951176425, 39988722875, 70920437325, 122735050305 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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.

EXAMPLE

First differences:    1, 63,  665,  3367, 11529, ... (A022522)

--------------------------------------------------------------------

The sixth powers:     1, 64,  729,  4096, 15625, ... (A001014)

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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)

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]

CROSSREFS

Cf. A000540, A001014, A022522, A101093, A254472, A254640, A254645, A254683, A254869, A254870, A254871.

Sequence in context: A289899 A267475 A241940 * A198449 A093271 A201608

Adjacent sequences:  A254869 A254870 A254871 * A254873 A254874 A254875

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 17 2015

STATUS

approved

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Last modified February 26 03:29 EST 2020. Contains 332273 sequences. (Running on oeis4.)