A199257
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 * x^k * A(x)^k]* x^n/n ).
Original entry on oeis.org
1, 1, 5, 18, 86, 408, 2075, 10787, 57655, 313643, 1733450, 9700574, 54867895, 313145033, 1801150861, 10430094658, 60758092753, 355795743385, 2093295146379, 12367548160650, 73346850194969, 436486017193373, 2605656191324094, 15599323024019360, 93634195155551584
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 86*x^4 + 408*x^5 + 2075*x^6 +...
such that A(x) = G(x*A(x)) where G(x) = (1-x+x^2)*(1+x^2)^2/(1-x)^2:
G(x) = 1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 12*x^5 + 16*x^6 + 20*x^7 + 24*x^8 +...
...
Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + 2^2*A + A^2)*x +
(1 + 4^2*A + 6^2*A^2 + 4^2*A^3 + A^4)*x^2/2 +
(1 + 6^2*A + 15^2*A^2 + 20^2*A^3 + 15^2*A^4 + 6^2*A^5 + A^6)*x^3/3 +
(1 + 8^2*A + 28^2*A^2 + 56^2*A^3 + 70^2*A^4 + 56^2*A^5 + 28^2*A^6 + 8^2*A^7 + A^8)*x^4/4 +
(1 + 10^2*A + 45^2*A^2 + 120^2*A^3 + 210^2*A^4 + 252^2*A^5 + 210^2*A^6 + 120^2*A^7 + 45^2*A^8 + 10^2*A^9 + A^10)*x^5/5 +...
which involves the squares of binomial coefficients C(2*n,k).
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{a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)^2/((1-x+x^2)*(1+x^2)^2+x*O(x^n))); polcoeff(A, n)}
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, binomial(2*m, k)^2 *x^k*A^k) *x^m/m)+x*O(x^n))); polcoeff(A, n)}
A198059
a(n) = Sum_{k=1..n} binomial(2*k, n-k)^2 * n/k.
Original entry on oeis.org
1, 9, 28, 121, 496, 2100, 9017, 38969, 169975, 744984, 3282005, 14513236, 64394500, 286519305, 1277975053, 5712392313, 25581765122, 114754116351, 515530099946, 2319115721576, 10445215621547, 47096725844837, 212569226371737, 960306310551860, 4341968468524371
Offset: 1
L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 121*x^4/4 + 496*x^5/5 + 2100*x^6/6 + ...
where
exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 52*x^4 + 187*x^5 + 708*x^6 + ... + A197601(n)*x^n + ...
The l.g.f. equals the series:
L(x) = (1 + 2^2*x + x^2)*x
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^2/2
+ (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^3/3
+ (1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)*x^4/4
+ (1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)*x^5/5 + ...
which involves the squares of the coefficients in even powers of (1+x).
Also,
L(x) = (1-x)^5*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 + ...)*x
+ (1-x)^9*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 + ...)*x^2/2
+ (1-x)^13*(1 + 7^2*x + 28^2*x^2 + 84^2*x^3 + 210^2*x^4 + ...)*x^3/3
+ (1-x)^17*(1 + 9^2*x + 45^2*x^2 + 165^2*x^3 + 495^2*x^4 + ...)*x^4/4
+ (1-x)^21*(1 + 11^2*x + 66^2*x^2 + 286^2*x^3 + 1001^2*x^4 + ...)*x^5/5 + ...
which involves the squares of the coefficients in odd powers of 1/(1-x).
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w := (x^3-2*x^2-3*x-1)*(x^3-2*x^2+5*x-1);
sqrt((1 + (15*x^3+2*x^2-x+3)/w - (x^3-2*x^2+x-4)/sqrt(w))/2) - 2;
series(%,x=0,30); # Mark van Hoeij, May 06 2013
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Table[Sum[Binomial[2k,n-k]^2 n/k,{k,n}],{n,30}] (* Harvey P. Dale, Oct 25 2011 *)
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{a(n)=n*sum(k=1,n,binomial(2*k,n-k)^2/k)}
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{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(2*m, k)^2 *x^k)*x^m/m)+x*O(x^n), n)}
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{a(n)=n*polcoeff(sum(m=1, n, (1-x+x*O(x^n))^(4*m+1) *sum(k=0, n-m+1, binomial(2*m+k, k)^2 *x^k)*x^m/m+x*O(x^n)), n)}
Showing 1-2 of 2 results.