A026800 Number of partitions of n in which the least part is 7.

0, 0, 0, 0, 0, 0, 0, 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

Offset

0,22

Comments

From Jason Kimberley, Feb 03 2011: (Start)

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the i-cycle; the addition of integers corresponds to the disconnected union of cycles.

By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End)

Links

Robert Israel, Table of n, a(n) for n = 0..10000

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

Formula

G.f.: x^7 * Product_{m>=7} 1/(1-x^m).

a(n) = p(n-7) -p(n-8) -p(n-9) +p(n-12) +2*p(n-14) -p(n-16) -p(n-17) -p(n-18) -p(n-19) +2*p(n-21) +p(n-23) -p(n-26) -p(n-27) +p(n-28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. - Shanzhen Gao, Oct 28 2010; offset corrected / made explicit by Jason Kimberley, Feb 03 2011

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>=1} x^(7*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(7)=1 partition is 7.
The a(14)=1 partition is 7+7.
The a(15)=1 partition is 7+8.
.............................
The a(20)=1 partition is 7+13.
The a(21)=2 partitions are 7+7+7 and 7+14.

Maple

N:= 100: # for a(0)..a(N)
S:= series(x^7/mul(1-x^i, i=7..N-7), x, N+1):
seq(coeff(S, x, i), i=0..N); # Robert Israel, Jul 04 2019

Mathematica

CoefficientList[Series[x^7/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)

Prog

(Magma) p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026800 := func< n | p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)- p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) >; // Jason Kimberley, Feb 03 2011
(Magma) R<x>:=PowerSeriesRing(Integers(), 75); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^7/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019
(PARI) my(x='x+O('x^75)); concat([0, 0, 0, 0, 0, 0, 0], Vec(x^7/prod(m=0, 80, 1-x^(m+7)))) \\ G. C. Greubel, Nov 03 2019
(Sage)
def A026800_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x^7/product((1-x^(m+7)) for m in (0..80)) ).list()
A026800_list(75) # G. C. Greubel, Nov 03 2019

Crossrefs

Cf. A185327 (Mathematica code)

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), A026799 (g=6), this sequence (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 03 2011

Sequence in context: A185229 A026825 A025150 * A185327 A210717 A171962

Adjacent sequences: A026797 A026798 A026799 * A026801 A026802 A026803

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

Status

approved