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A274099 Number of partitions of n*(n-1)/2 into at most four parts.

%I A274099 #19 Jul 08 2022 19:04:52
%S A274099 1,1,3,9,23,54,120,249,478,864,1495,2484,3969,6136,9234,13561,19464,
%T A274099 27378,37845,51488,69012,91260,119239,154078,197026,249535,313290,
%U A274099 390144,482120,591519,720954,873264,1051513,1259130,1499950,1778097,2097984,2464489
%N A274099 Number of partitions of n*(n-1)/2 into at most four parts.
%H A274099 Colin Barker, <a href="/A274099/b274099.txt">Table of n, a(n) for n = 1..1000</a>
%F A274099 Coefficient of x^(n*(n-1)/2) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%F A274099 Empirical g.f.: (1 -5*x +15*x^2 -30*x^3 +54*x^4 -77*x^5 +109*x^6 -128*x^7 +150*x^8 -148*x^9 +150*x^10 -128*x^11 +109*x^12 -77*x^13 +54*x^14 -30*x^15 +15*x^16 -5*x^17 +x^18) / ((1 -x)^7*(1 +x^2)^3*(1 +x +x^2)*(1 +x^4)). - _Colin Barker_, Jun 12 2016
%t A274099 Length[IntegerPartitions[#,4]]&/@Accumulate[Range[0,40]] (* _Harvey P. Dale_, Jul 08 2022 *)
%o A274099 (PARI)
%o A274099 \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%o A274099 b(n) = round(real((68+36*(-1)^n+18*((-I)^n+I^n)+(16*exp(-2/3*I*n*Pi)*(1+I*sqrt(3)+2*exp((4*I*n*Pi)/3)))/(1+(-1)^(1/3))+59*(1+n)+9*(-1)^n*(1+n)+18*(1+n)*(2+n)+2*(1+n)*(2+n)*(3+n))/288))
%o A274099 vector(50, n, b(n*(n-1)/2)) \\ _Colin Barker_, Jun 12 2016
%Y A274099 A subsequence of A001400.  Cf. A274100.
%K A274099 nonn
%O A274099 1,3
%A A274099 _N. J. A. Sloane_, Jun 11 2016
%E A274099 More terms from _Colin Barker_, Jun 12 2016