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),", "))
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),", "))
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 1-3 of 3 results.
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