A166991
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/(2*n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 1, 3, 12, 57, 300, 1693, 10045, 61890, 392688, 2550843, 16891566, 113660475, 775223595, 5349057132, 37280705406, 262119009927, 1857241951359, 13250054817027, 95110710932424, 686490953423700, 4979704242810870
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
log(A(x)^2) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
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a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(2*n), {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
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{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)/2*x^m/m)+x*O(x^n)),n)}
A218119
G.f.: A(x) = exp( Sum_{n>=1} A069865(n)*x^n/n ) where A069865(n) = Sum_{k=0..n} C(n,k)^6.
Original entry on oeis.org
1, 2, 35, 554, 15297, 451842, 15929824, 601077640, 24488754772, 1046792248856, 46718718597567, 2155032002133834, 102259392504591235, 4967499746642163574, 246231868462969357492, 12419324761881256326288, 635990044563649443993091, 33006906229799699591298070
Offset: 0
G.f.: A(x) = 1 + 2*x + 35*x^2 + 554*x^3 + 15297*x^4 + 451842*x^5 + 15929824*x^6 +...
log(A(x)) = 2*x + 66*x^2/2 + 1460*x^3/3 + 54850*x^4/4 + 2031252*x^5/5 + 86874564*x^6/6 + 3848298792*x^7/7 +...+ A069865(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^6)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))
A218118
G.f.: A(x) = exp( Sum_{n>=1} A005261(n)/2*x^n/n ) where A005261(n) = Sum_{k=0..n} C(n,k)^5.
Original entry on oeis.org
1, 1, 9, 90, 1350, 22623, 430338, 8786367, 190473510, 4314088755, 101271596421, 2446843690671, 60557118315384, 1529356193511525, 39297344717526330, 1024958399339092751, 27083985050402731646, 723942169622258974974, 19548657715769940178730
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 90*x^3 + 1350*x^4 + 22623*x^5 + 430338*x^6 +...
log(A(x)) = x + 17*x^2/2 + 244*x^3/3 + 4913*x^4/4 + 103126*x^5/5 + 2367152*x^6/6 + 56622784*x^7/7 +...+ A005261(n)/2*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^5)/2*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))
Showing 1-3 of 3 results.
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