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A026799
Number of partitions of n in which the least part is 6.
20
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151
OFFSET
0,19
COMMENTS
a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.
FORMULA
G.f.: x^6 * Product_{m>=6} 1/(1-x^m).
a(n) = p(n-6) -p(n-7) -p(n-8) +p(n-11) +p(n-12) +p(n-13) -p(n-14) -p(n-15) -p(n-16) +p(n-19) +p(n-20) -p(n-21) for n>0 where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(6*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020
EXAMPLE
a(0)=0 because there does not exist a least part of the empty partition.
The a(6)=1 partition is 6.
The a(12)=1 partition is 6+6.
The a(13)=1 partition is 6+7.
.............................
The a(17)=1 partition is 6+11.
The a(18)=2 partitions are 6+6+6 and 6+12.
MAPLE
ZL := [ B, {B=Set(Set(Z, card>=6))}, unlabeled ]: 0, 0, 0, 0, 0, 0, seq(combstruct[count](ZL, size=n), n=0..63); # Zerinvary Lajos, Mar 13 2007
seq(coeff(series(x^6/mul(1-x^(m+6), m=0..70), x, n+1), x, n), n = 0..65); # G. C. Greubel, Nov 03 2019
MATHEMATICA
f[1, 1]=f[0, k_]=1; f[n_, k_]:= f[n, k] = If[n<0, 0, If[k>n, 0, If[k==n, 1, f[n, k+1] +f[n-k, k]]]]; Join[{0, 0, 0, 0, 0, 0}, Table[f[n, 6], {n, 0, 65}]] (* Robert G. Wilson v, Jan 31 2011 *)
CoefficientList[Series[x^6/QPochhammer[x^6, x], {x, 0, 70}], x] (* G. C. Greubel, Nov 03 2019 *)
Join[{0}, Table[Count[IntegerPartitions[n][[;; , -1]], 6], {n, 70}]] (* Harvey P. Dale, Dec 27 2023 *)
PROG
(Magma) p := func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // Jason Kimberley, Feb 04 2011
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^6/(&*[1-x^(m+6): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019
(PARI) my(x='x+O('x^60)); concat([0, 0, 0, 0, 0, 0], Vec(x^6/prod(m=0, 70, 1-x^(m+6)))) \\ G. C. Greubel, Nov 03 2019
(Sage)
def A026799_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^6/product((1-x^(m+6)) for m in (0..70)) ).list()
A026799_list(65) # G. C. Greubel, Nov 03 2019
CROSSREFS
Essentially the same as A185326.
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), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 04 2011
Sequence in context: A185228 A026824 A025149 * A185326 A238209 A342096
KEYWORD
nonn,easy
STATUS
approved