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 (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
FORMULA
G.f.: (x + 11*x^2 + 11*x^3 + x^4)/(- 1 + x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400.
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^4.
EXAMPLE
Second differences: 2, 14, 50, 110, 194, 302, ... A120328(2k+1)
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, ... A254470
Seventh partial sums: 1, 23, 221, 1355, 6239, 23465, ... (this sequence)
MATHEMATICA
Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n)((7 + 42 n + 6 n^2)/19958400), {n, 24}] (* or *)
CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(- 1 + x)^12, {x, 0, 23}], x]
PROG
(PARI) vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400) \\ Derek Orr, Feb 19 2015
(Magma) [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+2*n)*(7 +42*n+6*n^2)/19958400: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Feb 17 2015
STATUS
approved