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A026797 Number of partitions of n in which the least part is 4. 23
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
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
G.f.: Sum_{k>=1} x^(4*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020
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<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)
def A026797_list(prec):
P.<x> = 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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified February 27 12:16 EST 2024. Contains 370375 sequences. (Running on oeis4.)