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)}
A216356
a(n) = A000172(n^2), where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 2, 346, 5280932, 6332299624282, 548057409594239814752, 3282684865686445066146128050420, 1329153351023643434414727317328867397924832, 35862023917618878200052422822926970148356592776600354650, 63875599229358329592315180101212796802405282289343043273094466311541144
Offset: 0
L.g.f.: L(x) = 2*x + 346*x^2/2 + 5280932*x^3/3 + 6332299624282*x^4/4 + 548057409594239814752*x^5/5 +...
where exp(L(x)) = 1 + 2*x + 175*x^2 + 1760658*x^3 + 1583078442003*x^4 + 109611485085305859618*x^5 +...+ A216355(n)*x^n +...
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{a(n)=sum(k=0, n^2, binomial(n^2, k)^3)}
for(n=0, 15, print1(a(n), ", "))
A383796
Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).
Original entry on oeis.org
1, 462, 396453, 425295010, 511915968714, 661059663660060, 895093835464198893, 1254056426977089876570, 1802794259810040618367902, 2644298823194748929633091780, 3941742074897786728895080586082, 5954164159064906497558129244865108, 9094122817144126105637193154022530612
Offset: 0
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seq(n)=Vec(exp(sum(n=1, n, (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!), O(x*x^n)))) \\ Andrew Howroyd, Jun 11 2025
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)}
A362731
a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ).
Original entry on oeis.org
1, 2, 18, 182, 1954, 21702, 246366, 2839846, 33105186, 389264798, 4608481918, 54862022910, 656099844526, 7876525155020, 94867757934870, 1145843922848232, 13873839714404642, 168345900709550388, 2046612356962697502, 24923311881995950740, 303974276349311203854
Offset: 0
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A000172 := proc(n) add(binomial(n,k)^3, k = 0..n); end:
E(n,x) := series( exp(n*add(A000172(k)*x^k/k, k = 1..20)), x, 21 ):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
A384957
Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).
Original entry on oeis.org
1, 990, 2206149, 6450139410, 21553605027306, 77957908218716988, 297118041166459732781, 1175248212459867447863562, 4779368947089383238327733950, 19858241947988743766121587718308, 83936671517628352407663509802203682, 359778601391313651280693986124971038388, 1560159110515342136997114532804454280500084
Offset: 0
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seq(n)=Vec(exp(sum(n=1, n, (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n, O(x*x^n)) )) \\ Andrew Howroyd, Jun 13 2025
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