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),", "))
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),", "))
A158266
G.f.: A(x) = exp( Sum_{n>=1} C(2n-1,n)^2*x^n/n ).
Original entry on oeis.org
1, 1, 5, 38, 352, 3659, 41012, 484739, 5959417, 75523708, 980470867, 12980840984, 174675568464, 2382923659387, 32890264803521, 458576476085929, 6450351908991558, 91437202854436755, 1305115286958337403
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 352*x^4 + 3659*x^5 + 41012*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 10^2*x^3/3 + 35^2*x^4/4 + 126^2*x^5/5 +...
log(C(x)) = x + 3*x^2/2 + 10*x^3/3 + 35*x^4/4 + 126*x^5/5 +...
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 +... (g.f. of A000108).
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{a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m-1,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-1,k)^2*a(n-k)))}
A362730
a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} binomial(2*k,k)^2*x^k/k ).
Original entry on oeis.org
1, 4, 68, 1336, 27972, 607004, 13478072, 304083224, 6941422916, 159882680452, 3708781743068, 86526900550864, 2028273983776440, 47733938489878528, 1127187050415921304, 26694934151138897336, 633813403549444601156, 15082008687681962081088, 359592614152718921447108
Offset: 0
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E(n,x) := series( exp(n*add(binomial(2*k,k)^2*x^k/k, k = 1..20)), x, 21 ):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
Showing 1-5 of 5 results.
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