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A254646 Fourth partial sums of seventh powers (A001015). 6
1, 132, 2709, 26432, 168126, 804552, 3136014, 10459968, 30856839, 82407052, 202678203, 465069696, 1005729452, 2066218896, 4058958828, 7664805504, 13974953853, 24692818836, 42415687153, 71020845504 (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 (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

FORMULA

G.f.: (x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7)/(- 1 + x)^12.

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(48 - 100*n - 89*n^2 + 160*n^3 + 140*n^4 + 36*n^5 + 3*n^6)/23760.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^7.

EXAMPLE

First differences:   1, 127, 2059, 14197,  61741, ...  (A022523)

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

The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)

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

First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)

Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)

Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)

Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (this sequence)

MATHEMATICA

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (48 - 100 n - 89 n^2 + 160 n^3 + 140 n^4 + 36 n^5 + 3 n^6)/23760, {n, 20}] (* or *)

Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^7]]]] (* or *)

CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(- 1 + x)^12, {x, 0, 19}], x]

PROG

(PARI) a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(48-100*n-89*n^2+160*n^3+140*n^4+36*n^5+3*n^6)/23760 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A000541, A001015, A022523, A250212, A254641, A254645.

Sequence in context: A064303 A220925 A220996 * A242147 A023902 A240270

Adjacent sequences:  A254643 A254644 A254645 * A254647 A254648 A254649

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 05 2015

STATUS

approved

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Last modified May 21 15:32 EDT 2019. Contains 323444 sequences. (Running on oeis4.)