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A254682 Fifth partial sums of fifth powers (A000584). 8
1, 37, 418, 2754, 13080, 49632, 159654, 452166, 1157013, 2724865, 5988268, 12410476, 24456744, 46132152, 83740980, 146935284, 250134753, 414416277, 669990046, 1059399550, 1641605680, 2497140360, 3734542890, 5498322570 (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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

FORMULA

G.f.: (- x - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^11.

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(- 2 + 5*n + n^2)*(9 + 10*n + 2*n^2)/60480.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^5.

EXAMPLE

Fifth differences:   1, 27,  93,  119,   120, (repeat 120) (A101100)

Fourth differences:  1, 28, 121,  240,   360,   480, ...   (A101095)

Third differences:   1, 29, 150,  390,   750,  1230, ...   (A101096)

Second differences:  1, 30, 180,  570,  1320,  2550, ...   (A101098)

First differences:   1, 31, 211,  781,  2101,  4651, ...   (A022521)

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

The fifth powers:    1, 32, 243, 1024,  3125,  7776, ...   (A000584)

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

First partial sums:  1, 33, 276, 1300,  4425, 12201, ...   (A000539)

Second partial sums: 1, 34, 310, 1610,  6035, 18236, ...   (A101092)

Third partial sums:  1, 35, 345, 1955,  7990, 26226, ...   (A101099)

Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ...   (A254644)

Fifth partial sums:  1, 37, 418, 2754, 13080, 49632, ...   (this sequence)

MATHEMATICA

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 2 + 5 n + n^2) (9 + 10 n + 2 n^2)/60480, {n, 24}] (* or *)

CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^11, {x, 0, 23}], x]

Nest[Accumulate, Range[30]^5, 5] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 37, 418, 2754, 13080, 49632, 159654, 452166, 1157013, 2724865, 5988268}, 30] (* Harvey P. Dale, Jan 30 2019 *)

PROG

(PARI) a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-2+5*n+n^2)*(9+10*n+2*n^2)/60480 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A000539, A000584, A022521, A101092, A101095, A101096, A101098, A101099, A101100, A254644, A254681, A254683, A254684.

Sequence in context: A115926 A083818 A090023 * A232251 A232259 A232046

Adjacent sequences:  A254679 A254680 A254681 * A254683 A254684 A254685

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 12 2015

STATUS

approved

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Last modified January 26 05:10 EST 2020. Contains 331273 sequences. (Running on oeis4.)