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.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
G.f.: (-x - 11*x^2 - 11*x^3 - x^4)/(- 1 + x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(1 + 12*n + 2*n^2)/302400.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^4.
Sum_{n>=1} 1/a(n) = 3320303/2601 + 1400*Pi^2/17 + (8960/17)*sqrt(2/17)*Pi*cot(sqrt(17/2)*Pi). - Amiram Eldar, Jan 26 2022
EXAMPLE
First differences: 1, 15, 65, 175, 369, 671, ... (A005917)
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The fourth powers: 1, 16, 81, 256, 625, 1296, ... (A000583)
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First partial sums: 1, 17, 98, 354, 979, 2275, ... (A000538)
Second partial sums: 1, 18, 116, 470, 1449, 3724, ... (A101089)
Third partial sums: 1, 19, 135, 605, 2054, 5778, ... (A101090)
Fourth partial sums: 1, 20, 155, 760, 2814, 8592, ... (A101091)
Fifth partial sums: 1, 21, 176, 936, 3750, 12342, ... (A254681)
Sixth partial sums: 1, 22, 198,1134, 4884, 17226, ... (this sequence)
MATHEMATICA
Table[n (1 + n) (2 + n) (3 + n)^2 (4 + n) (5 + n) (6 + n) (1 + 12 n + 2 n^2)/302400, {n, 25}] (* or *) CoefficientList[Series[(- 1 - 11 x - 11 x^2 - x^3)/(- 1 + x)^11, {x, 0, 24}], x]
Nest[Accumulate, Range[30]^4, 6] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 22, 198, 1134, 4884, 17226, 52338, 141570, 348777, 795652, 1701700}, 30] (* Harvey P. Dale, Apr 23 2016 *)
PROG
(Magma) [n*(1+n)*(2+n)*(3+n)^2*(4+n)*(5+n)*(6+n)*(1+12*n+ 2*n^2)/302400: n in [1..30]]; // Vincenzo Librandi, Feb 15 2015
(PARI) vector(50, n, n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(1 + 12*n + 2*n^2)/302400) \\ Derek Orr, Feb 19 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Feb 15 2015
STATUS
approved