A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

0, 0, 0, 0, 0, 0, 1, 2, 4, 3, 7, 8, 11, 11, 17, 17, 23, 23, 30, 31, 39, 38, 48, 49, 58, 58, 70, 70, 82, 82, 95, 96, 110, 109, 125, 126, 141, 141, 159, 159, 177, 177, 196, 197, 217, 216, 238, 239, 260, 260, 284, 284, 308, 308, 333, 334, 360, 359, 387, 388, 415, 415, 445

Offset: 1

Author

Clark Kimberling, Jan 01 2007

Keywords

• nonn
• easy

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)

Examples

			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)
		

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)

Crossrefs

Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.

Programs

Magma

R:=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

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 *)

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

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

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

a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - R. J. Mathar, Nov 23 2013