A299043 G.f. Sum_{n>=0} Series_Reversion( x*(1-x)^n )^n.

1, 1, 2, 7, 33, 191, 1293, 9941, 85137, 801067, 8194281, 90367696, 1067146336, 13418399528, 178808377777, 2514944176091, 37204969293137, 577131827509491, 9362170099804501, 158438822236836110, 2791230865213193695, 51090157185364462103, 969892719975254406849, 19066076629590290124814, 387539455534509836620517, 8134022943287699194376826, 176073319016203896275830713

Offset: 0

Author

Paul D. Hanna, Feb 18 2018

Keywords

• nonn

Examples

			G.f. A(x) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 191*x^5 + 1293*x^6 + 9941*x^7 + 85137*x^8 + 801067*x^9 + 8194281*x^10 + ...
such that
A(x) = 1 + x*R(x,2) + x^2*R(x,3)^4 + x^3*R(x,4)^9 + x^4*R(x,5)^16 + x^5*R(x,6)^25 + x^6*R(x,7)^36 + ...
where series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
and series R(x,n+1)^(n^2) begin:
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ...
R(x,3)^4 = 1 + 4*x + 18*x^2 + 88*x^3 + 455*x^4 + 2448*x^5 + ...
R(x,4)^9 = 1 + 9*x + 72*x^2 + 570*x^3 + 4554*x^4 + 36855*x^5 + ...
R(x,5)^16 = 1 + 16*x + 200*x^2 + 2320*x^3 + 26180*x^4 + 292448*x^5 + ...
R(x,6)^25 = 1 + 25*x + 450*x^2 + 7175*x^3 + 108100*x^4 + 1581255*x^5 + ...
R(x,7)^36 = 1 + 36*x + 882*x^2 + 18480*x^3 + 357399*x^4 + 6601644*x^5 + ...
...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A299044.

Programs

PARI

{a(n) = my(A); A = sum(m=0,n+1, serreverse( x*(1-x)^m +x^2*O(x^n) )^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

PARI

{a(n) = if(n==0,1, sum(k=0,n, binomial(n*(n-k) + k,k) * (n-k)^2/(n*(n-k) + k) ) )}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following expressions.

(1) A(x) = Sum_{n>=0} Series_Reversion( x*(1-x)^n )^n.

(2) A(x) = Sum_{n>=1} x^n * R(x,n+1)^(n^2), where

(2.a) R(x,n+1) = 1 + x*R(x,n+1)^(n+1),

(2.b) R(x,n+1)^n = Series_Reversion( x*(1-x)^n ) / x,

(2.c) R(x,n+1)^n = Sum_{k>=0} C(n*(k+1) + k, k) * n/(n*(k+1) + k) * x^k,

(2.d) R(x,n+1)^(n^2) = Sum_{k>=0} C(n*(n+k) + k, k) * n^2/(n*(n+k) + k) * x^k.

FORMULAS FOR TERMS.

a(n) = Sum_{k=0..n} binomial(n*(n-k) + k, k) * (n-k)^2/(n*(n-k) + k).