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)}
A166992
G.f.: A(x) = exp( Sum_{n>=1} A005260(n)*x^n/n ) where A005260(n) = Sum_{k=0..n} C(n,k)^4.
Original entry on oeis.org
1, 2, 11, 74, 621, 5850, 60212, 659712, 7583514, 90494068, 1112755389, 14022849582, 180362150901, 2360201899690, 31344689243344, 421621652965160, 5734850816825046, 78773961705345324, 1091497852618784390
Offset: 0
G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 621*x^4 + 5850*x^5 + 60212*x^6 +...
log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 1810*x^4/4 + 21252*x^5/5 + 263844*x^6/6 + 3395016*x^7/7 +...+ A005260(n)*x^n/n +...
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a[n_] := Sum[(Binomial[n, k])^4, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(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)^4)*x^m/m)+x*O(x^n)),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),", "))
A218120
G.f.: A(x) = exp( Sum_{n>=1} A069865(n)/2*x^n/n ) where A069865(n) = Sum_{k=0..n} C(n,k)^6.
Original entry on oeis.org
1, 1, 17, 260, 7244, 214257, 7593707, 287419304, 11745920475, 503237634257, 22503750152879, 1039694201489294, 49401095274561608, 2402478324494963930, 119201977436336120482, 6017223412990713126034, 308361587173800754305214, 16013543997544827365960598
Offset: 0
G.f.: A(x) = 1 + x + 17*x^2 + 260*x^3 + 7244*x^4 + 214257*x^5 + 7593707*x^6 +...
log(A(x)) = x + 33*x^2/2 + 730*x^3/3 + 27425*x^4/4 + 1015626*x^5/5 + 43437282*x^6/6 + 1924149396*x^7/7 +...+ A069865(n)/2*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^6)/2*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))
Showing 1-4 of 4 results.
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