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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. Sum_{n>=1} 1/a(n) = 1920*sqrt(3/7)*Pi*tan(sqrt(21)*Pi/2) - 251488/49. - Amiram Eldar, Jan 26 2022 EXAMPLE 2nd differences: 0, 6, 12, 18, 24, 30, ... (A008588) 1st differences: 1, 7, 19, 37, 61, 91, ... (A003215) ------------------------------------------------------------------- The cubes: 1, 8, 27, 64, 125, 216, ... (A000578) ------------------------------------------------------------------- 1st partial sums: 1, 9, 36, 100, 225, 441, ... (A000537) 2nd partial sums: 1, 10, 46, 146, 371, 812, ... (A024166) 3rd partial sums: 1, 11, 57, 203, 574, 1386, ... (A101094) 4th partial sums: 1, 12, 69, 272, 846, 2232, ... (A101097) 5th partial sums: 1, 13, 82, 354, 1200, 3432, ... (A101102) 6th partial sums: 1, 14, 96, 450, 1650, 5082, ... (A254469) 7th 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 April 19 08:20 EDT 2024. Contains 371782 sequences. (Running on oeis4.)