A198944
G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k] * x^n/n ).
Original entry on oeis.org
1, 1, 2, 7, 23, 78, 291, 1126, 4436, 17910, 73773, 308188, 1303402, 5573133, 24050795, 104620985, 458324429, 2020417339, 8956142180, 39899217350, 178549985024, 802275736073, 3618237414959, 16373514195570, 74325340129430, 338356926399193, 1544406450870590
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 23*x^4 + 78*x^5 + 291*x^6 +...
where
log(A(x)) = (1 + x*A(x))*x + (1 + 2^3*x*A(x) + x^2*A(x)^2)*x^2/2 +
(1 + 3^3*x*A(x) + 3^3*x^2*A(x)^2 + x^3*A(x)^3)*x^3/3 +
(1 + 4^3*x*A(x) + 6^3*x^2*A(x)^2 + 4^3*x^3*A(x)^3 + x^4*A(x)^4)*x^4/4 +
(1 + 5^3*x*A(x) + 10^3*x^2*A(x)^2 + 10^3*x^3*A(x)^3 + 5^3*x^4*A(x)^4 + x^5*A(x)^5)*x^5/5 +...
more explicitly,
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 63*x^4/4 + 251*x^5/5 + 1110*x^6/6 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*(x*A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}
A200212
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^(n-k)] * x^n/n ).
Original entry on oeis.org
1, 1, 3, 11, 42, 174, 763, 3457, 16075, 76351, 368767, 1805682, 8943948, 44736096, 225646033, 1146461185, 5862224756, 30144922281, 155791900727, 808773877919, 4215675455503, 22054576750972, 115765182718467, 609508331610920, 3218059655553030, 17034314889643633
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 174*x^5 + 763*x^6 +...
where the logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (A + x)*x + (A^2 + 2^3*x*A + x^2)*x^2/2 +
(A^3 + 3^3*x*A^2 + 3^3*x^2*A + x^3)*x^3/3 +
(A^4 + 4^3*x*A^3 + 6^3*x^2*A^2 + 4^3*x^3*A + x^4)*x^4/4 +
(A^5 + 5^3*x*A^4 + 10^3*x^2*A^3 + 10^3*x^3*A^2 + 5^3*x^4*A + x^5)*x^5/5 +
(A^6 + 6^3*x*A^5 + 15^3*x^2*A^4 + 20^3*x^3*A^3 + 15^3*x^4*A^2 + 6^3*x^5*A + x^6)*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 117*x^4/4 + 581*x^5/5 + 2987*x^6/6 + 15499*x^7/7 + 81213*x^8/8 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
A192204
G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^4*A(x)^k) * x^n/n ).
Original entry on oeis.org
1, 2, 13, 109, 1099, 12283, 147620, 1869346, 24633344, 334916467, 4669887745, 66481991644, 963096090267, 14160279233964, 210870471771803, 3175275874056722, 48281516978747396, 740504452581897112, 11444972742343813815
Offset: 0
G.f.: A(x) = 1 + 2*x + 13*x^2 + 109*x^3 + 1099*x^4 + 12283*x^5 +...
which satisfies:
log(A(x)) = (1 + A(x))*x + (1 + 16*A(x) + A(x)^2)*x^2/2 + (1 + 81*A(x) + 81*A(x)^2 + A(x)^3)*x^3/3 + (1 + 256*A(x) + 1296*A(x)^2 + 256*A(x)^3 + A(x)^4)*x^4/4 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^4*(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}
A200215
G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k) * x^n*A(x)^n/n ).
Original entry on oeis.org
1, 1, 3, 13, 61, 306, 1623, 8937, 50565, 292283, 1718827, 10250916, 61854848, 376949934, 2316738789, 14343701657, 89379109846, 560108223900, 3527723269978, 22318890516413, 141778326349191, 903936594232782, 5782447430948438, 37102633354583532, 238729798670985104
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 61*x^4 + 306*x^5 + 1623*x^6 +...
where the logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A)*x*A + (1 + 2^3*x*A + x^2*A^2)*x^2*A^2/2 +
(1 + 3^3*x*A + 3^3*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +
(1 + 4^3*x*A + 6^3*x^2*A^2 + 4^3*x^3*A^3 + x^4*A^4)*x^4*A^4/4 +
(1 + 5^3*x*A + 10^3*x^2*A^2 + 10^3*x^3*A^3 + 5^3*x^4*A^4 + x^5*A^5)*x^5*A^5/5 +
(1 + 6^3*x*A + 15^3*x^2*A^2 + 20^3*x^3*A^3 + 15^3*x^4*A^4 + 6^3*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 185*x^4/4 + 1126*x^5/5 + 7043*x^6/6 + 44689*x^7/7 + 286241*x^8/8 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j*A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
Showing 1-4 of 4 results.
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