A140092 G.f. satisfies: A(x) = Series_Reversion[ x/sqrt(1 + 4*A(x)) ] with A(0)=0.

1, 2, 6, 28, 174, 1308, 11300, 108808, 1145078, 12996332, 157580252, 2026874424, 27507762028, 392226116696, 5855551243464, 91263899531280, 1481385005886374, 24989341719984972, 437270678940944556, 7923785627972483672

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Formula

G.f. satisfies: A(x) = x*sqrt(1 + 4*A(A(x))).

G.f. satisfies: A(A(x)) = [(A(x)^2 - x^2]/(2*x)^2.

a(n)=T(n,1), T(n, m)=m/n*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*sum(j=0..i-k, binomial(-k-j+2*i-1,i-1)*(k+j)*2^(k+j)*binomial(k+j-1,k-1))/i*(-1)^(i+k))*binomial(n+k-1, n-1)), n>m, T(n,n)=1. [Vladimir Kruchinin, May 08 2012]

Example

G.f.: A(x) = x +2*x^2 +6*x^3 + 28*x^4 + 174*x^5 +1308*x^6 +11300*x^7 +...
A(A(x)) = x + 4*x^2 + 20*x^3 +124*x^4 + 912*x^5 +7676*x^6 +72064*x^7 +...
A(x)^2 = x^2 +4*x^3 + 16*x^4 + 80*x^5 + 496*x^6 +3648*x^7 +30704*x^8 +...

Prog

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x/sqrt(1+4*A +x*O(x^n)))); polcoeff(A, n))}
(Maxima) array(TL, fixnum, 30, 30); T(n, m):=if n=m then 1 else if TL[n, m]=0 then TL[n, m]:m/n*sum(sum(T(n-m, i)*sum(binomial(-k-j+2*i-1, i-1)*(k+j)*2^(k+j)*binomial(k+j-1, k-1), j, 0, i-k)/i*(-1)^(i+k), i, k, n-m)*binomial(n+k-1, n-1), k, 1, n-m) else TL[n, m]; makelist(T(n, 1), n, 1, 15); [Vladimir Kruchinin, May 08 2012]

Crossrefs

Cf. A087949.

Sequence in context: A156626 A207386 A088501 * A052809 A136631 A002435

Adjacent sequences: A140089 A140090 A140091 * A140093 A140094 A140095

Keyword

nonn

Author

Paul D. Hanna, May 15 2008

Status

approved