A127002 - OEIS

A127002

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

%I #35 Oct 21 2022 22:03:11

%S 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,

%T 70,82,82,95,96,110,109,125,126,141,141,159,159,177,177,196,197,217,

%U 216,238,239,260,260,284,284,308,308,333,334,360,359,387,388,415,415,445

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

%C From _Gus Wiseman_, Apr 19 2019: (Start)

%C 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:

%C (21) (31) (32) (42) (43) (53) (54)

%C (211) (41) (51) (52) (62) (63)

%C (221) (411) (61) (71) (72)

%C (311) (322) (332) (81)

%C (331) (422) (441)

%C (511) (611) (522)

%C (3211) (3221) (711)

%C (4211) (3321)

%C (4221)

%C (4311)

%C (5211)

%C (End)

%H G. C. Greubel, <a href="/A127002/b127002.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,-1,-1,-1,0,1)

%F G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - _Vladeta Jovovic_, Jan 03 2007

%F 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

%F a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - _Mircea Merca_, Nov 23 2013

%F a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - _R. J. Mathar_, Nov 23 2013

%e a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.

%e 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}

%e From _Gus Wiseman_, Apr 19 2019: (Start)

%e 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.

%e (3211) (3221) (3321) (5221) (4322) (4332) (4432)

%e (4211) (4221) (5311) (4331) (4431) (5332)

%e (4311) (6211) (4421) (5322) (5422)

%e (5211) (5411) (5331) (5521)

%e (6221) (6411) (6322)

%e (6311) (7221) (6331)

%e (7211) (7311) (6511)

%e (8211) (7411)

%e (8221)

%e (8311)

%e (9211)

%e (End)

%p 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

%p 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:

%p 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:

%p for n from 1 to 200 do print(A127002(n)) ; od ; # _R. J. Mathar_, Jan 07 2007

%t Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* _Gus Wiseman_, Apr 19 2019 *)

%t 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 *)

%o (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

%o (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

%o (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

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

%K nonn,easy

%O 1,8

%A _Clark Kimberling_, Jan 01 2007