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A299043 G.f. Sum_{n>=0} Series_Reversion( x*(1-x)^n )^n. 2

%I #8 Feb 18 2018 14:57:16

%S 1,1,2,7,33,191,1293,9941,85137,801067,8194281,90367696,1067146336,

%T 13418399528,178808377777,2514944176091,37204969293137,

%U 577131827509491,9362170099804501,158438822236836110,2791230865213193695,51090157185364462103,969892719975254406849,19066076629590290124814,387539455534509836620517,8134022943287699194376826,176073319016203896275830713

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

%H Paul D. Hanna, <a href="/A299043/b299043.txt">Table of n, a(n) for n = 0..500</a>

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

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

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

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

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

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

%F (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.

%F FORMULAS FOR TERMS.

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

%e 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 + ...

%e such that

%e 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 + ...

%e where series R(x,n) = 1 + x*R(x,n)^n begin:

%e R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...

%e R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...

%e R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...

%e R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...

%e R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...

%e R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...

%e ...

%e and series R(x,n+1)^(n^2) begin:

%e R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ...

%e R(x,3)^4 = 1 + 4*x + 18*x^2 + 88*x^3 + 455*x^4 + 2448*x^5 + ...

%e R(x,4)^9 = 1 + 9*x + 72*x^2 + 570*x^3 + 4554*x^4 + 36855*x^5 + ...

%e R(x,5)^16 = 1 + 16*x + 200*x^2 + 2320*x^3 + 26180*x^4 + 292448*x^5 + ...

%e R(x,6)^25 = 1 + 25*x + 450*x^2 + 7175*x^3 + 108100*x^4 + 1581255*x^5 + ...

%e R(x,7)^36 = 1 + 36*x + 882*x^2 + 18480*x^3 + 357399*x^4 + 6601644*x^5 + ...

%e ...

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = if(n==0,1, sum(k=0,n, binomial(n*(n-k) + k,k) * (n-k)^2/(n*(n-k) + k) ) )}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A299044.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 18 2018

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