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A185327
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Number of partitions of n into parts >= 7.
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20
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1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045
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OFFSET
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0,15
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COMMENTS
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a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 7 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. Hence this sequence is essentially the same as A026800.
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LINKS
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FORMULA
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G.f.: Product_{m>=7} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + 2*p(n-7) - p(n-9) - p(n-10) - p(n-11) - p(n-12) + 2*p(n-14) + p(n-16) - p(n-19) - p(n-20) + p(n-21) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010 [moved/copied from A026800 by Jason Kimberley, Feb 03 2011]
This sequence is the Euler transformation of A185117.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(7*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
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EXAMPLE
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The a(0)=1 empty partition vacuously has each part >= 7.
The a(7)=1 partition is 7.
The a(8)=1 partition is 8.
............................
The a(13)=1 partition is 13.
The a(14)=2 partitions are 7+7 and 14.
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MAPLE
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seq(coeff(series(1/mul(1-x^(m+7), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019
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MATHEMATICA
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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]]]]; Table[ f[n, 7], {n, 0, 65}] (* Robert G. Wilson v, Jan 31 2011 *) (* moved from A026800 by Jason Kimberley, Feb 03 2011 *)
Join[{1}, Table[Count[IntegerPartitions[n], _?(Min[#]>=7&)], {n, 0, 70}]] (* Harvey P. Dale, Oct 16 2011 *)
CoefficientList[Series[1/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)
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PROG
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(Magma) p := func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185327 := func< n | p(n)-p(n-1)-p(n-2)+p(n-5)+2*p(n-7)-p(n-9)-p(n-10)- p(n-11)-p(n-12)+2*p(n-14)+p(n-16)-p(n-19)-p(n-20)+p(n-21) >;
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019
(PARI) my(x='x+O('x^70)); Vec(1/prod(m=0, 80, 1-x^(m+7))) \\ G. C. Greubel, Nov 03 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/product((1-x^(m+7)) for m in (0..80)) ).list()
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CROSSREFS
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2-regular simple graphs with girth at least 7: A185117 (connected), A185227 (disconnected), this sequence (not necessarily connected).
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), this sequence (g=7), A185328 (g=8), A185329 (g=9).
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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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