A201556
G.f.: exp( Sum_{n>=1} C(2*n^2,n^2) * x^n/n ).
Original entry on oeis.org
1, 2, 37, 16278, 150303194, 25282422428664, 73752140616074524401, 3639659041645240391812731402, 2993893262520330875797362908273443346, 40656420461436928818704580402413441308206341488, 9054851465691640957562090101797213977192016103053025996396
Offset: 0
G.f.: A(x) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 +...
where
log(A(x)) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 +...+ C(2*n^2,n^2)*x^n/n +...
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nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Mar 06 2014 *)
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{a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m^2,m^2)*x^m/m)+x*O(x^n)),n)}
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{a(n)=if(n==0,1,(1/n)*sum(k=1,n,binomial(2*k^2,k^2)*a(n-k)))}
A224732
G.f.: exp( Sum_{n>=1} binomial(2*n,n)^n * x^n/n ).
Original entry on oeis.org
1, 2, 20, 2704, 6008032, 203263062688, 103724721990326528, 801185400238209125917312, 94088900962948953837864576996352, 168691065596220817138271126002845218561536, 4634314586972355372645450331391809316221983940020224
Offset: 0
G.f.: A(x) = 1 + 2*x + 20*x^2 + 2704*x^3 + 6008032*x^4 + 203263062688*x^5 +...
where
log(A(x)) = 2*x + 6^2*x^2/2 + 20^3*x^3/3 + 70^4*x^4/4 + 252^5*x^5/5 + 924^6*x^6/6 + 3432^7*x^7/7 + 12870^8*x^8/8 +...+ A000984(n)^n*x^n/n +...
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{a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^k*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
A213402
Expansion of exp( Sum_{n>=1} binomial(2*n^2-1, n^2) * x^n/n ).
Original entry on oeis.org
1, 1, 18, 8121, 75143314, 12641135924840, 36876057665515782388, 1819829483946334379705083199, 1496946629440335290077827171482478552, 20328210229221517747154723779733726877327901902, 4527425732825492268524878479039840102127714838348518156649
Offset: 0
G.f.: A(x) = 1 + x + 18*x^2 + 8121*x^3 + 75143314*x^4 + 12641135924840*x^5 +...
where
log(A(x)) = x + 35*x^2/2 + 24310*x^3/3 + 300540195*x^4/4 +...+ C(2*n^2-1,n^2)*x^n/n +...
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nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]/2*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Mar 06 2014 *)
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{a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m^2, m^2)/2*x^m/m)+x*O(x^n)), n)}
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{a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(2*k^2, k^2)/2*a(n-k)))}
for(n=0,21,print1(a(n),", "))
A213410
G.f.: exp( Sum_{n>=1} binomial(3*n,n)^n/3^n * x^n/n ).
Original entry on oeis.org
1, 1, 13, 7330, 185307558, 201002187396362, 9357300769149011773697, 18775362849239140086719414696830, 1631039199744298058694966065590003308698494, 6159916689356522044764167426829149420348399496664634288
Offset: 0
G.f.: A(x) = 1 + x + 13*x^2 + 7330*x^3 + 185307558*x^4 + 201002187396362*x^5 +...
where
log(A(x)) = x + 5^2*x^2/2 + 28^3*x^3/3 + 165^4*x^4/4 + 1001^5*x^5/5 + 6188^6*x^6/6 + 38760^7*x^7/7 +...+ A025174(n)^n*x^n/n +...
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nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k,k]^k/3^k * b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Mar 06 2014 *)
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{a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m, m)^m/3^m*x^m/m)+x*O(x^n)), n)}
for(n=0,15,print1(a(n),", "))
Showing 1-4 of 4 results.
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