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A026803
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Number of partitions of n in which the least part is 10.
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18
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0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 27, 29, 34, 37, 43, 47, 54, 59, 68, 74, 85, 93, 106, 116, 132, 145, 164, 180, 203, 223, 252, 276, 310, 341, 382, 420, 470, 516, 576, 633, 706, 775, 863
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OFFSET
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1,30
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COMMENTS
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In general, if g>=1 and g.f. = x^g * Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018
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LINKS
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FORMULA
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G.f.: x^10 * Product_{m>=10} 1/(1-x^m).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*sqrt(2)*Pi^9 / (3*n^(11/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(10*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020
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MAPLE
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seq(coeff(series(x^10/mul(1-x^(m+10), m = 0..85), x, n+1), x, n), n = 1..80); # G. C. Greubel, Nov 03 2019
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MATHEMATICA
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Rest@CoefficientList[Series[x^10/QPochhammer[x^10, x], {x, 0, 80}], x] (* G. C. Greubel, Nov 03 2019 *)
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PROG
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(PARI) my(x='x+O('x^80)); concat(vector(9), Vec(x^10/prod(m=0, 85, 1-x^(m+10)))) \\ G. C. Greubel, Nov 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 80); [0, 0, 0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^10/(&*[1-x^(m+10): m in [0..85]]) )); // G. C. Greubel, Nov 03 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^10/product((1-x^(m+10)) for m in (0..85)) ).list()
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
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STATUS
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approved
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