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A254869 Seventh partial sums of cubes (A000578). 5
1, 15, 111, 561, 2211, 7293, 21021, 54483, 129558, 286858, 598026, 1184118, 2242266, 4083366, 7184166, 12257850, 20348031, 32951985, 52179985, 80958735, 123288165, 184562235, 271965915, 394962165, 565884540, 800652996, 1119632580, 1548656956 (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*(1 + 4*x + x^2)/(1 - x)^11.

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800.

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^3.

EXAMPLE

Second differences:   0,  6,  12,   18,   24,   30, ... (A008588)

First differences:    1,  7,  19,   37,   61,   91, ... (A003215)

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

The cubes:            1,  8,  27,   64,  125,  216, ... (A000578)

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

First partial sums:   1,  9,  36,  100,  225,  441, ... (A000537)

Second partial sums:  1, 10,  46,  146,  371,  812, ... (A024166)

Third partial sums:   1, 11,  57,  203,  574, 1386, ... (A101094)

Fourth partial sums:  1, 12,  69,  272,  846, 2232, ... (A101097)

Fifth partial sums:   1, 13,  82,  354, 1200, 3432, ... (A101102)

Sixth partial sums:   1, 14,  96,  450, 1650, 5082, ... (A254469)

Seventh partial sums: 1, 15, 111,  561, 2211, 7293, ... (this sequence)

MATHEMATICA

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 7 n + n^2)/604800, {n, 26}] (* or *)

CoefficientList[Series[(- 1 - 4 x - x^2)/(- 1 + x)^11, {x, 0, 25}], x]

Nest[Accumulate, Range[30]^3, 7] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 15, 111, 561, 2211, 7293, 21021, 54483, 129558, 286858, 598026}, 30] (* Harvey P. Dale, Apr 24 2017 *)

PROG

(PARI) vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800) \\ Derek Orr, Feb 19 2015

(MAGMA) [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+7*n+n^2)/604800: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015

CROSSREFS

Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254469, A254870, A254871, A254872.

Sequence in context: A290752 A290753 A290361 * A034184 A092646 A222117

Adjacent sequences:  A254866 A254867 A254868 * A254870 A254871 A254872

KEYWORD

nonn,easy

AUTHOR

Luciano Ancora, Feb 17 2015

STATUS

approved

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Last modified May 27 06:24 EDT 2019. Contains 323599 sequences. (Running on oeis4.)