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 (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
G.f.:(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^10.
a(n) = n^2*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(5 + 2*n)/30240.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^4.
E.g.f.: (1/30240)*exp(x)*(30240 + 604800*x + 2041200*x^2 + 2368800*x^3 + 1233540*x^4 + 326592*x^5 + 46410*x^6 + 3540*x^7 + 135*x^8 + 2*x^9). - Stefano Spezia, Dec 02 2018
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 172032*log(2)/125 - 2382233/2500.
Sum_{n>=1} (-1)^(n+1)/a(n) = 42*Pi^2/25 - 43008*Pi/125 + 2663213/2500. (End)
EXAMPLE
Fourth differences: 1, 12, 23, 24, (repeat 24) ... (A101104)
Third differences: 1, 13, 36, 60, 84, 108, ... (A101103)
Second differences: 1, 14, 50, 110, 194, 302, ... (A005914)
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, ... (this sequence)
MAPLE
seq(coeff(series((x+11*x^2+11*x^3+x^4)/(1-x)^10, x, n+1), x, n), n = 1 .. 30); # Muniru A Asiru, Dec 02 2018
MATHEMATICA
Table[n^2(1+n)(2+n)(3+n)(4+n)(5+n)^2(5+2n)/30240, {n, 26}] (* or *)
CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(1-x)^10, {x, 0, 25}], x]
CoefficientList[Series[(1/30240)E^x (30240 + 604800 x + 2041200 x^2 + 2368800 x^3 + 1233540 x^4 + 326592 x^5 + 46410 x^6 + 3540 x^7 + 135 x^8 + 2 x^9), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 02 2018 *)
Nest[Accumulate[#]&, Range[30]^4, 5] (* Harvey P. Dale, Jan 03 2022 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((x+11*x^2+11*x^3+x^4)/(1-x)^10) \\ G. C. Greubel, Dec 01 2018
(Magma) [Binomial(n+5, 6)*n*(n+5)*(2*n+5)/42: n in [1..30]]; // G. C. Greubel, Dec 01 2018
(Sage) [binomial(n+5, 6)*n*(n+5)*(2*n+5)/42 for n in (1..30)] # G. C. Greubel, Dec 01 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Feb 12 2015
STATUS
approved