A219228
O.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(x)).
Original entry on oeis.org
1, 1, 32, 3119, 625710, 214333471, 112105268136, 83149960883200, 83014425998481126, 107334569041127441462, 174471878478682785998864, 348242875992753988109552778, 837327855535084109106340786272, 2387108242583316451939303856237037
Offset: 0
O.g.f.: A(x) = 1 + x + 32*x^2 + 3119*x^3 + 625710*x^4 + 214333471*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(x)^2/2!*exp(-8*x*A(x)) + 3^9*x^3*A(x)^3/3!*exp(-27*x*A(x)) + 4^12*x^4*A(x)^4/4!*exp(-64*x*A(x)) + 5^15*x^5*A(x)^5/5!*exp(-125*x*A(x)) +...
simplifies to a power series in x with integer coefficients.
G.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
G(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + 210766920*x^5 + 110687251039*x^6 +...+ Stirling2(3*n,n)*x^n +...
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{a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^3*x*A)^m/m!*exp(-m^3*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))
A219264
O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n*x)).
Original entry on oeis.org
1, 1, 8, 128, 3259, 120082, 6151625, 433404057, 42180568185, 5720993700540, 1088246094845838, 291276119631119408, 109983236494820652007, 58741463418913578672779, 44466318283501559718838424, 47771843216826858235974983400, 72930986725295232949801895385998
Offset: 0
O.g.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3259*x^4 + 120082*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2*x)^2/2!*exp(-2^2*x*A(2*x)) + 3^6*x^3*A(3*x)^3/3!*exp(-3^2*x*A(3*x)) + 4^8*x^4*A(4*x)^4/4!*exp(-4^2*x*A(4*x)) + 5^10*x^5*A(5*x)^5/5!*exp(-5^2*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(2*k)*x^k*subst(A,x,k*x)^k/k!*exp(-k^2*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A219265
O.g.f. satisfies: A(x) = Sum_{n>=0} A(n^2*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n^2*x)).
Original entry on oeis.org
1, 1, 8, 160, 6918, 609469, 106947753, 37651271215, 26931993643529, 39243099256414069, 116654228928308598913, 710224935200206160129234, 8867331728829780268501045551, 227187317486051730833557991305666, 11969414396907448200529521385052444890
Offset: 0
O.g.f.: A(x) = 1 + x + 8*x^2 + 160*x^3 + 6918*x^4 + 609469*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2^2*x)^2/2!*exp(-2^2*x*A(2^2*x)) + 3^6*x^3*A(3^2*x)^3/3!*exp(-3^2*x*A(3^2*x)) + 4^8*x^4*A(4^2*x)^4/4!*exp(-4^2*x*A(4^2*x)) + 5^10*x^5*A(5^2*x)^5/5!*exp(-5^2*x*A(5^2*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(2*k)*x^k*subst(A,x,k^2*x)^k/k!*exp(-k^2*x*subst(A,x,k^2*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A219343
O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(n*x)).
Original entry on oeis.org
1, 1, 32, 3183, 650929, 226009218, 119298668857, 89086101638412, 89480710389500666, 116491795770107486363, 191172400354899371561288, 387419202671209086703674709, 956322827450633453264262285623, 2859815748552720894795327258080881, 10430012061189048036456303441601971435
Offset: 0
O.g.f.: A(x) = 1 + x + 32*x^2 + 3183*x^3 + 650929*x^4 + 226009218*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(2*x)^2/2!*exp(-2^3*x*A(2*x)) + 3^9*x^3*A(3*x)^3/3!*exp(-3^3*x*A(3*x)) + 4^12*x^4*A(4*x)^4/4!*exp(-4^3*x*A(4*x)) + 5^15*x^5*A(5*x)^5/5!*exp(-5^3*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(3*k)*x^k*subst(A,x,k*x)^k/k!*exp(-k^3*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
Showing 1-4 of 4 results.
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