This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127002 #35 Oct 21 2022 22:03:11
%S A127002 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 A127002 70,82,82,95,96,110,109,125,126,141,141,159,159,177,177,196,197,217,
%U A127002 216,238,239,260,260,284,284,308,308,333,334,360,359,387,388,415,415,445
%N A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.
%C A127002 From _Gus Wiseman_, Apr 19 2019: (Start)
%C A127002 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 A127002 (21) (31) (32) (42) (43) (53) (54)
%C A127002 (211) (41) (51) (52) (62) (63)
%C A127002 (221) (411) (61) (71) (72)
%C A127002 (311) (322) (332) (81)
%C A127002 (331) (422) (441)
%C A127002 (511) (611) (522)
%C A127002 (3211) (3221) (711)
%C A127002 (4211) (3321)
%C A127002 (4221)
%C A127002 (4311)
%C A127002 (5211)
%C A127002 (End)
%H A127002 G. C. Greubel, <a href="/A127002/b127002.txt">Table of n, a(n) for n = 1..1000</a>
%H A127002 <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 A127002 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 A127002 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 A127002 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 A127002 a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - _R. J. Mathar_, Nov 23 2013
%e A127002 a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
%e A127002 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 A127002 From _Gus Wiseman_, Apr 19 2019: (Start)
%e A127002 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 A127002 (3211) (3221) (3321) (5221) (4322) (4332) (4432)
%e A127002 (4211) (4221) (5311) (4331) (4431) (5332)
%e A127002 (4311) (6211) (4421) (5322) (5422)
%e A127002 (5211) (5411) (5331) (5521)
%e A127002 (6221) (6411) (6322)
%e A127002 (6311) (7221) (6331)
%e A127002 (7211) (7311) (6511)
%e A127002 (8211) (7411)
%e A127002 (8221)
%e A127002 (8311)
%e A127002 (9211)
%e A127002 (End)
%p A127002 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 A127002 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 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 A127002 for n from 1 to 200 do print(A127002(n)) ; od ; # _R. J. Mathar_, Jan 07 2007
%t A127002 Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* _Gus Wiseman_, Apr 19 2019 *)
%t A127002 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 A127002 (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 A127002 (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 A127002 (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 A127002 Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.
%K A127002 nonn,easy
%O A127002 1,8
%A A127002 _Clark Kimberling_, Jan 01 2007