A046919 Maximal coefficient of polynomial p(n), with p(3)=1, p(n) = (1 - t^(2*n - 4))*(1 - t^(2*n - 3))*p(n - 1)/((1 - t^(n - 3))*(1 - t^n)).

1, 1, 3, 8, 24, 73, 227, 734, 2430, 8150, 27718, 95514, 332578, 1168261, 4136477, 14749992, 52925886, 190973410, 692583902, 2523265494, 9231352260, 33901898722, 124940568222, 461938289518, 1713007181342, 6369928427268, 23747917426918, 88747514693530, 332397792962692, 1247582980566935, 4691740496135919, 17676678143316236, 66714895880626460, 252207367615436780

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

a(n) is also the number of partitions of n(n-1)/2 into n (nonzero) parts, none greater than n-2 [Riordan].

Examples

			1; 1+t+t^2+t^3+t^4+t^5, t^10+t^9+2*t^8+2*t^7+3*t^6+3*t^5+3*t^4+2*t^3+2*t^2+t+1, ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 3..200 (terms 3..50 from N. J. A. Sloane)
• J. Riordan, The number of score sequences in tournaments, J. Combin. Theory, 5 (1968), 87-89. [The main result of this paper seems to be wrong - compare A000571 and A210726.]

Crossrefs

Cf. A000571, A046918, A210726.

Programs

Maple

p := proc(n)
option remember;
if n = 3 then 1 else
simplify((1-t^(2*n-4))*(1-t^(2*n-3))*p(n-1)/((1-t^(n-3))*(1-t^n)));
fi; end;
for i from 3 to 40 do
lprint(coeff(expand(p(i)),t,i*(i-3)/2)):
od:

Mathematica

p[3] = 1; p[n_] := p[n] = (1 - t^(2*n-4))*(1 - t^(2*n-3))*(p[n-1]/((1 - t^(n-3))*(1 - t^n)))// Simplify // Expand; a[n_] := Coefficient[p[n], t, n*(n-3)/2]; Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Aug 01 2013, after Maple *)

Formula

a(n) ~ sqrt(3) * 2^(2*n-3) / (Pi * n^2). - Vaclav Kotesovec, Jan 07 2023

Extensions

Corrected terms and Maple program. - N. J. A. Sloane, May 09 2012