A224734
G.f.: exp( Sum_{n>=1} binomial(2*n,n)^2 * x^n/n ).
Original entry on oeis.org
1, 4, 26, 216, 2075, 21916, 247326, 2930216, 36028117, 456089076, 5910983050, 78100285784, 1048696065394, 14275198859304, 196610207633100, 2735542102308752, 38400942393884068, 543307627503591440, 7740605626606127512, 110970838624540461472, 1599834676405793089013
Offset: 0
G.f.: A(x) = 1 + 4*x + 26*x^2 + 216*x^3 + 2075*x^4 + 21916*x^5 + 247326*x^6 +...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 20^2*x^3/3 + 70^2*x^4/4 + 252^2*x^5/5 + 924^2*x^6/6 + 3432^2*x^7/7 + 12870^2*x^8/8 +...+ A000984(n)^2*x^n/n +...
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CoefficientList[Series[Exp[4*x*HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 27 2025 *)
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{a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^2*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
A200002
G.f.: exp( Sum_{n>=1} C(2*n,n)^n/2^n * x^n/n ).
Original entry on oeis.org
1, 1, 5, 338, 375502, 6351970709, 1620698781098852, 6259260939361008796229, 367534769386519350929158503892, 329474737492618783473185792974307067503, 4525697838840190793599072589249813785373031191426, 955617474162634862818320009634143510233705849191099879550608
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 338*x^3 + 375502*x^4 + 6351970709*x^5 +...
where
log(A(x)) = x + 3^2*x^2/2 + 10^3*x^3/3 + 35^4*x^4/4 + 126^5*x^5/5 + 462^6*x^6/6 + 1716^7*x^7/7 +...+ A001700(n+1)^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[2*k,k]^k/2^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(2*m,m)^m/2^m*x^m/m)+x*O(x^n)),n)}
A224733
a(n) = binomial(2*n,n)^n.
Original entry on oeis.org
1, 2, 36, 8000, 24010000, 1016255020032, 622345892187672576, 5608296349498479967469568, 752711194884611945703392100000000, 1518219588672387021538193329290752000000000, 46343145866349732399475841723454160331675252923826176
Offset: 0
L.g.f.: L(x) = 2*x + 36*x^2/2 + 8000*x^3/3 + 24010000*x^4/4 + 1016255020032*x^5/5 +...
Equivalently,
L(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 +...
where exponentiation yields an integer series:
exp(L(x)) = 1 + 2*x + 20*x^2 + 2704*x^3 + 6008032*x^4 + 203263062688*x^5 + 103724721990326528*x^6 +...+ A224732(n)*x^n +...
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Table[Binomial[2n,n]^n,{n,0,10}] (* Harvey P. Dale, Apr 19 2016 *)
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{a(n)=binomial(2*n,n)^n}
for(n=0,20,print1(a(n),", "))
A224735
G.f.: exp( Sum_{n>=1} binomial(2*n,n)^3 * x^n/n ).
Original entry on oeis.org
1, 8, 140, 3616, 116542, 4316080, 175593800, 7640774080, 349626142909, 16632958651688, 816163494236860, 41069537125459360, 2110206360805542510, 110346590629125981872, 5857345961837113457864, 314962180518584299711424, 17128125582951726423704502, 940726748732537798295599280
Offset: 0
G.f.: A(x) = 1 + 8*x + 140*x^2 + 3616*x^3 + 116542*x^4 + 4316080*x^5 +...
where
log(A(x)) = 2^3*x + 6^3*x^2/2 + 20^3*x^3/3 + 70^3*x^4/4 + 252^3*x^5/5 + 924^3*x^6/6 + 3432^3*x^7/7 + 12870^3*x^8/8 +...+ A000984(n)^3*x^n/n +...
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CoefficientList[Series[Exp[8*x*HypergeometricPFQ[{1, 1, 3/2, 3/2, 3/2}, {2, 2, 2, 2}, 64*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 27 2025 *)
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{a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^3*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
A224736
G.f.: exp( Sum_{n>=1} binomial(2*n,n)^4 * x^n/n ).
Original entry on oeis.org
1, 16, 776, 64384, 7151460, 947788608, 141137282720, 22814994697728, 3918995299504938, 705339416079749024, 131725296229995045840, 25348575698532710671104, 5000341179482293108254824, 1007144334380887781805059200, 206487157000689985136888031296
Offset: 0
G.f.: A(x) = 1 + 16*x + 776*x^2 + 64384*x^3 + 7151460*x^4 + 947788608*x^5 +...
where
log(A(x)) = 2^4*x + 6^4*x^2/2 + 20^4*x^3/3 + 70^4*x^4/4 + 252^4*x^5/5 + 924^4*x^6/6 + 3432^4*x^7/7 + 12870^4*x^8/8 +...+ A000984(n)^4*x^n/n +...
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CoefficientList[Series[Exp[16*x*HypergeometricPFQ[{1, 1, 3/2, 3/2, 3/2, 3/2}, {2, 2, 2, 2, 2}, 256*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 27 2025 *)
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{a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^4*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
Showing 1-5 of 5 results.
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