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A008484 Number of partitions of n into parts >= 4. 27
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, 4443, 5113, 5834, 6698 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 4 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Jan 2011 and Feb 2012

By removing a single part of size 4, an A026797 partition of n becomes an A008484 partition of n - 4. Hence this sequence is essentially the same as A026797. - Jason Kimberley, Feb 2012

Number of partitions of n+3 such that 3*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

Let c(n) be the number of partitions of n such that both (number of parts) and 2*(number of parts) are parts; then c(n) = a(n-6) for n >= 6 and c(n) = 0 for n < 6. - Clark Kimberling, Mar 01 2014

a(n) is also the number of partitions of n for which three times the number of ones is twice the number of parts (conjectured). - George Beck, Aug 19 2017

LINKS

Giovanni Resta, Table of n, a(n) for n = 0..1000

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4

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

FORMULA

G.f.: 1 / Product_{m>=4} (1 - x^m).

Euler transformation of A185114. - Jason Kimberley, Jan 30 2011

Given by p(n) - p(n-1) - p(n-2) + p(n-4) + p(n-5) - p(n-6) where p(n) = A000041(n). Generally, 1/Product_{i>=K} (1 - x^i) is given by p({A}), where {A} is defined over the coefficients of Product_{i=1..K-1} (1 - x^i). In this case, K=4, so (1-x)(1-x^2)(1-x^3) = 1 - x - x^2 + x^4 + x^5 - x^6, defining {A} as above. G.f.: 1 + Sum_{i>=1} (x^4i)/Product_{j=1..i}(1 - x^j). - Jon Perry, Jul 04 2004

MAPLE

series(1/product((1-x^i), i=4..50), x, 51); # end of program

ZL := [ B, {B=Set(Set(Z, card>=4))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..49); # Zerinvary Lajos, Mar 13 2007

MATHEMATICA

f[1, 1] = 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, 4], {n, 49}] (* end of program *)

Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Length[p]]], {n, 40}], 2]  (* Clark Kimberling, Feb 27 2014 *)

Table[Count[IntegerPartitions[n],

  p_ /; 3 Count[p, 1] == 2 Length[p]], {n, 0, 50}] (* George Beck Aug 19 2017 *)

PROG

(MAGMA) a:= func< n | NumberOfPartitions(n)-NumberOfPartitions(n-1)-NumberOfPartitions(n-2)+ NumberOfPartitions(n-4)+NumberOfPartitions(n-5)- NumberOfPartitions(n-6) >; [1, 0, 0, 0, 1, 1, 1] cat [ a(n) : n in [7..100]]; // Vincenzo Librandi, Aug 20 2017

CROSSREFS

2-regular graphs with girth at least 4: A185114 (connected), A185224 (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), this sequence (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), A026796 (g=3), A026797 (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 with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: this sequence (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).

Sequence in context: A126793 A069910 A026797 * A274146 A027189 A140829

Adjacent sequences:  A008481 A008482 A008483 * A008485 A008486 A008487

KEYWORD

nonn,easy

AUTHOR

T. Forbes (anthony.d.forbes(AT)googlemail.com)

STATUS

approved

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Last modified November 20 00:42 EST 2017. Contains 294957 sequences.