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

1, 1, 3, 9, 23, 54, 120, 249, 478, 864, 1495, 2484, 3969, 6136, 9234, 13561, 19464, 27378, 37845, 51488, 69012, 91260, 119239, 154078, 197026, 249535, 313290, 390144, 482120, 591519, 720954, 873264, 1051513, 1259130, 1499950, 1778097, 2097984, 2464489

Offset: 1

Author

N. J. A. Sloane, Jun 11 2016

Keywords

• nonn

Links

• Colin Barker, Table of n, a(n) for n = 1..1000

Crossrefs

A subsequence of A001400. Cf. A274100.

Programs

Mathematica

Length[IntegerPartitions[#,4]]&/@Accumulate[Range[0,40]] (* Harvey P. Dale, Jul 08 2022 *)

PARI

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
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))
vector(50, n, b(n*(n-1)/2)) \\ Colin Barker, Jun 12 2016

Formula

Coefficient of x^(n*(n-1)/2) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

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

Extensions

More terms from Colin Barker, Jun 12 2016