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A026801
Number of partitions of n in which the least part is 8.
19
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 21, 23, 27, 30, 36, 39, 46, 51, 60, 66, 77, 85, 99, 110, 126, 140, 162, 179, 205, 228, 260, 289, 329, 365, 415, 461, 521, 579, 655, 726, 818, 909, 1022, 1134, 1273, 1411
OFFSET
1,24
FORMULA
G.f.: x^8 * Product_{m>=8} 1/(1-x^m).
a(n+8) = p(n) -p(n-1) -p(n-2) +p(n-5) +p(n-7) +p(n-8) -p(n-10) -p(n-11) -2*p(n-12) +2*p(n-16) +p(n-17) +p(n-18) -p(n-20) -p(n-21) -p(n-23) +p(n-26) +p(n-27) -p(n-28) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(8*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020
MAPLE
seq(coeff(series(x^8/mul(1-x^(m+8), m = 0..80), x, n+1), x, n), n = 1..70); # G. C. Greubel, Nov 03 2019
MATHEMATICA
Rest@CoefficientList[Series[x^8/QPochhammer[x^8, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)
PROG
(PARI) my(x='x+O('x^70)); concat(vector(7), Vec(x^8/prod(m=0, 80, 1-x^(m+8)))) \\ G. C. Greubel, Nov 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^8/(&*[1-x^(m+8): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019
(Sage)
def A026801_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^8/product((1-x^(m+8)) for m in (0..80)) ).list()
a=A026801_list(71); a[1:] # G. C. Greubel, Nov 03 2019
CROSSREFS
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), this sequence (g=8), A026802 (g=9), A026803 (g=10).
Sequence in context: A264591 A026826 A025151 * A185328 A210718 A027191
KEYWORD
nonn,easy
EXTENSIONS
More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
STATUS
approved