OFFSET
1,8
COMMENTS
From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n - 4 of the form a+b, a+a+b, or a+a+b+c, ignoring ordering. A bijection can be constructed from the partitions described in the name by subtracting one from all parts and deleting zeros. These are also partitions with adjusted frequency depth (A323014, A325280) equal to their length plus one, and their Heinz numbers are given by A325281. For example, the a(7) = 1 through a(13) = 11 partitions are:
(21) (31) (32) (42) (43) (53) (54)
(211) (41) (51) (52) (62) (63)
(221) (411) (61) (71) (72)
(311) (322) (332) (81)
(331) (422) (441)
(511) (611) (522)
(3211) (3221) (711)
(4211) (3321)
(4221)
(4311)
(5211)
(End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1)
FORMULA
G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - Vladeta Jovovic, Jan 03 2007
G.f.: Sum_{k>=3} Sum_{j=2..k-1} Sum_{m=1..j-1} x^(m+j+k)*(x^m +x^j +x^k). - Emeric Deutsch, Jan 05 2007
a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - Mircea Merca, Nov 23 2013
EXAMPLE
a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
a(11) counts {1,1,2,7}, {1,1,3,6}, {1,1,4,5}, {2,2,1,6}, {2,2,3,4}, {3,3,1,4}, {4,4,1,2}
From Gus Wiseman, Apr 19 2019: (Start)
The a(7) = 1 through a(13) = 11 partitions of the form a+a+b+c are the following. The Heinz numbers of these partitions are given by A085987.
(3211) (3221) (3321) (5221) (4322) (4332) (4432)
(4211) (4221) (5311) (4331) (4431) (5332)
(4311) (6211) (4421) (5322) (5422)
(5211) (5411) (5331) (5521)
(6221) (6411) (6322)
(6311) (7221) (6331)
(7211) (7311) (6511)
(8211) (7411)
(8221)
(8311)
(9211)
(End)
MAPLE
g:=sum(sum(sum(x^(i+j+k)*(x^i+x^j+x^k), i=1..j-1), j=2..k-1), k=3..80): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=1..65); # Emeric Deutsch, Jan 05 2007
isA127002 := proc(p) local s; if nops(p) = 4 then s := convert(p, set) ; if nops(s) = 3 then RETURN(1) ; else RETURN(0) ; fi ; else RETURN(0) ; fi ; end:
A127002 := proc(n) local part, res, p; part := combinat[partition](n) ; res := 0 ; for p from 1 to nops(part) do res := res+isA127002(op(p, part)) ; od ; RETURN(res) ; end:
for n from 1 to 200 do print(A127002(n)) ; od ; # R. J. Mathar, Jan 07 2007
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]=={1, 1, 2}&]], {n, 70}] (* Gus Wiseman, Apr 19 2019 *)
Rest[CoefficientList[Series[x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)), {x, 0, 70}], x]] (* G. C. Greubel, May 30 2019 *)
PROG
(PARI) my(x='x+O('x^70)); concat(vector(6), Vec(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)))) \\ G. C. Greubel, May 30 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); [0, 0, 0, 0, 0, 0] cat Coefficients(R!( x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, May 30 2019
(Sage) a=(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 30 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 01 2007
STATUS
approved