A026797 Number of partitions of n in which the least part is 4.

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

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

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

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

Adjacent sequences: A026794 A026795 A026796 * A026798 A026799 A026800

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved