A304788
Expansion of e.g.f. exp(Sum_{k>=1} binomial(2*k,k)*x^k/(k + 1)!).
Original entry on oeis.org
1, 1, 3, 12, 59, 343, 2295, 17307, 144751, 1326377, 13189945, 141271298, 1619488645, 19766050827, 255693112641, 3492065507376, 50180426293255, 756444290843433, 11930511611596861, 196404976143077964, 3367697323914503113, 60029614473492823771, 1110430594720934758781
Offset: 0
E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 12*x^3/3! + 59*x^4/4! + 343*x^5/5! + 2295*x^6/6! + 17307*x^7/7! + ...
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a:=series(exp(add(binomial(2*k,k)*x^k/(k+1)!,k=1..100)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Exp[Sum[CatalanNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[CatalanNumber[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
A323667
Expansion of e.g.f. exp(BesselI(0,2*x) + BesselI(1,2*x) - 1).
Original entry on oeis.org
1, 1, 3, 10, 43, 211, 1191, 7463, 51535, 386809, 3133273, 27184620, 251253157, 2461527511, 25459020289, 276987375642, 3160197122183, 37705878268985, 469340324930493, 6081394853597162, 81866045488063721, 1142928276326927223, 16521454311961005245, 246917508673451732077
Offset: 0
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seq(n!*coeff(series(exp(BesselI(0,2*x)+BesselI(1,2*x)-1),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 28 2019
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nmax = 23; CoefficientList[Series[Exp[BesselI[0, 2 x] + BesselI[1, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
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my(x='x + O('x^25)); Vec(serlaplace(exp(besseli(0, 2*x)+x*besseli(1, 2*x)-1))) \\ Michel Marcus, Jan 24 2019
A323668
Expansion of e.g.f. exp(exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - 1).
Original entry on oeis.org
1, 3, 19, 152, 1467, 16445, 208471, 2934321, 45254447, 756995131, 13623709401, 262067291106, 5358900661509, 115953603121881, 2644399031839729, 63346390393538780, 1589177904965680263, 41642328796769014811, 1137083068108603968349, 32287430515011314674632, 951565685429585731747913
Offset: 0
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seq(n!*coeff(series(exp(exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))-1),x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 28 2019
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nmax = 20; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] + BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[2 k + 1, k + 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]
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my(x='x + O('x^25)); Vec(serlaplace(exp(exp(2*x)*(besseli(0, 2*x)+x*besseli(1, 2*x))-1))) \\ Michel Marcus, Jan 24 2019
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