A216354
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^n*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 2, 52, 58640, 3583098592, 11584364000042912, 2042518153012624794424576, 20047892010468651075834167466942080, 11138509206681372983092694151616405935206616064, 354938139483847646086359348765071470756626699510545192807936
Offset: 0
G.f.: A(x) = 1 + 2*x + 52*x^2 + 58640*x^3 + 3583098592*x^4 +...
where
log(A(x)) = 2*x + 10^2*x^2/2 + 56^3*x^3/3 + 346^4*x^4/4 + 2252^5*x^5/5 + 15184^6*x^6/6 + 104960^7*x^7/7 +...+ A000172(n)^n*x^n/n +...
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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)^m*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
A216352
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^2*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 4, 58, 1256, 35771, 1200188, 45016678, 1827941560, 78753548245, 3551810922324, 166120394053698, 8002733850225288, 395089619067741926, 19911864121386482264, 1021345223473335336668, 53190166903606336969840, 2807000233813092463820488, 149869216802426305919295328
Offset: 0
G.f.: A(x) = 1 + 4*x + 58*x^2 + 1256*x^3 + 35771*x^4 + 1200188*x^5 +...
such that
log(A(x)) = 4*x + 100*x^2/2 + 3136*x^3/3 + 119716*x^4/4 + 5071504*x^5/5 +...+ A000172(n)^2*x^n/n +...
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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)^2*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
A216353
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^3*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 8, 532, 62624, 10964914, 2399234384, 609215149096, 171739556144192, 52316948995446679, 16918106849112020088, 5736533516906891508780, 2021549577502367744673888, 735516733692051220039803750, 274907827442478316252748869104, 105138174536582510069969443280760
Offset: 0
G.f.: A(x) = 1 + 8*x + 532*x^2 + 62624*x^3 + 10964914*x^4 + 2399234384*x^5 +...
where
log(A(x)) = 2^3*x + 10^3*x^2/2 + 56^3*x^3/3 + 346^3*x^4/4 + 2252^3*x^5/5 + 15184^3*x^6/6 + 104960^3*x^7/7 +...+ A000172(n)^3*x^n/n +...
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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)^3*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
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), ", "))
Showing 1-4 of 4 results.