A166990
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 2, 7, 30, 147, 786, 4472, 26644, 164477, 1044258, 6782484, 44887236, 301782361, 2056250570, 14172792355, 98667874038, 692948001906, 4904403499992, 34951124337300, 250617829087656, 1807055528439771, 13095146839953030
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
log(A(x)) = 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/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
nmax = 30; Clear[a]; franel = RecurrenceTable[{n^2*a[n] == (7*n^2 - 7*n + 2)*a[n-1] + 8*(n-1)^2*a[n-2], a[1] == 2, a[2] == 10}, a, {n, 1, nmax}]; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[franel[[k]]*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 27 2024 *)
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{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)*x^m/m)+x*O(x^n)),n)}
A166993
G.f.: A(x) = exp( Sum_{n>=1} A005260(n)*x^n/(2*n) ), where A005260(n) = Sum_{k=0..n} C(n,k)^4.
Original entry on oeis.org
1, 1, 5, 32, 266, 2499, 25765, 283084, 3264502, 39077898, 481942608, 6089941550, 78523226064, 1029859481949, 13704960309415, 184688556173542, 2516342539576510, 34617557176739174, 480336524752492608
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 32*x^3 + 266*x^4 + 2499*x^5 + 25765*x^6 +...
log(A(x)) = x + 9*x^2/2 + 82*x^3/3 + 905*x^4/4 + 10626*x^5/5 + 131922*x^6/6 + 1697508*x^7/7 +...+ A005260(n)/2*x^n/n +...
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a[n_] := Sum[(Binomial[n, k])^4, {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)^4)/2*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),", "))
A199816
G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n)/4 * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).
Original entry on oeis.org
1, 1, 8, 101, 1639, 30665, 630225, 13836981, 319062453, 7640441894, 188534274850, 4767113222750, 122998902095908, 3228067183537455, 85960229675478804, 2317956019913480326, 63193008693741620771, 1739473925024629613227, 48292271242981605779173
Offset: 0
G.f.: A(x) = 1 + x + 8*x^2 + 101*x^3 + 1639*x^4 + 30665*x^5 +...
where
log(A(x)) = 1*1*x + 3*5*x^2/2 + 10*28*x^3/3 + 35*173*x^4/4 + 126*1126*x^5/5 + 462*7592*x^6/6 +...+ A000984(n)/2*A000172(n)/2*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m, m)/2*sum(k=0, m, binomial(m, k)^3)/2*x^m/m)+x*O(x^n)), n)}
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