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 A026797 Number of partitions of n in which the least part is 4. 22
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,12 COMMENTS a(n) is also the number of, not necessarily connected, 2-regular simple graphs girth exactly 4. - Jason Kimberley, Feb 22 2013 LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 FORMULA 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 MAPLE 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 MATHEMATICA 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 *) PROG (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:=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) def A026797_list(prec):     P. = PowerSeriesRing(ZZ, prec)     return P( x^4/product((1-x^(m+4)) for m in (0..60)) ).list() a=A026797_list(60); a[1:] # G. C. Greubel, Nov 03 2019 CROSSREFS Essentially the same as A008484. 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), 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). Sequence in context: A238789 A126793 A069910 * A008484 A274146 A027189 Adjacent sequences:  A026794 A026795 A026796 * A026798 A026799 A026800 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified February 23 11:21 EST 2020. Contains 332159 sequences. (Running on oeis4.)