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A026797
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Number of partitions of n in which the least part is 4.
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23
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0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 24, 27, 34, 39, 50, 57, 70, 81, 100, 115, 140, 161, 195, 225, 269, 311, 371, 427, 505, 583, 688, 791, 928, 1067, 1248, 1434, 1668, 1914, 2223, 2546, 2945, 3370, 3889
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OFFSET
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1,12
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COMMENTS
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a(n) is also the number of, not necessarily connected, 2-regular simple graphs girth exactly 4. - Jason Kimberley, Feb 22 2013
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LINKS
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FORMULA
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G.f.: x^4 * Product_{m>=4} 1/(1-x^m).
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(4*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^4/mul(1-x^(m+4), m=0..65), x, n+1), x, n), n = 1..60); # G. C. Greubel, Nov 03 2019
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MATHEMATICA
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Table[Count[IntegerPartitions[n], _?(Min[#]==4&)], {n, 60}] (* Harvey P. Dale, May 13 2012 *)
Rest@CoefficientList[Series[x^4/QPochhammer[x^4, x], {x, 0, 60}], x] (* G. C. Greubel, Nov 03 2019 *)
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PROG
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(PARI) my(x='x+O('x^60)); concat([0, 0, 0], Vec(x^4/prod(m=0, 70, 1-x^(m+4)))) \\ G. C. Greubel, Nov 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0, 0, 0] cat Coefficients(R!( x^4/(&*[1-x^(m+4): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^4/product((1-x^(m+4)) for m in (0..60)) ).list()
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CROSSREFS
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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), this sequence (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), A185644 (triangle); fixed k: this sequence (k=2), A185134 (k=3), A185144 (k=4).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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