A361065
E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).
Original entry on oeis.org
1, 1, 7, 85, 1521, 36421, 1097743, 39968601, 1707558401, 83777885929, 4643185678551, 286930307457949, 19562851003118833, 1458832806486727725, 118121195050068075167, 10320576944751955718881, 967863775658734350214017, 96970880819175875321264209
Offset: 0
-
A361065 := proc(n)
add((2*k+1)^(k-1)*binomial(n-1,n-k)/k!,k=0..n) ;
%*n! ;
end proc:
seq(A361065(n),n=0..10) ; # R. J. Mathar, Mar 02 2023
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nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
-
a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x))/2)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-(1-x)/(2*x)*lambertw(-2*x/(1-x)))))
A361066
E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).
Original entry on oeis.org
1, 1, 9, 148, 3673, 123276, 5234599, 269262022, 16279709793, 1131627199816, 88926737901031, 7796168316687906, 754414052156289265, 79872584117422215484, 9184299004593618881655, 1139822558262829096519726, 151857077047173825979147969
Offset: 0
-
nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
-
a(n) = n!*sum(k=0, n, (3*k+1)^(k-1)*binomial(n-1, n-k)/k!);
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x/(1-x))/3)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((-(1-x)/(3*x)*lambertw(-3*x/(1-x)))^(1/3)))
A361068
E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).
Original entry on oeis.org
1, 1, -1, 13, -127, 2101, -41801, 1030177, -29820127, 995977801, -37660751569, 1590847310581, -74242656468575, 3793664894534269, -210656932372422745, 12630986901470435401, -813335155262348743231, 55977540398642247218449
Offset: 0
-
nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^2)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
-
a(n) = n!*sum(k=0, n, (-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x/(1-x))/2)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt((1-x)/(2*x)*lambertw(2*x/(1-x)))))
A361069
E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).
Original entry on oeis.org
1, 1, -3, 40, -719, 18396, -598157, 23713726, -1108701519, 59735988424, -3644505746549, 248358786667674, -18697767289462967, 1541202721786228060, -138046868771541971373, 13351368704222195975206, -1386710317839048140282783, 153939247458296219191539984
Offset: 0
-
nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^3)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
-
a(n) = n!*sum(k=0, n, (-3*k+1)^(k-1)*binomial(n-1, n-k)/k!);
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*x/(1-x))/3)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/((1-x)/(3*x)*lambertw(3*x/(1-x)))^(1/3)))
A361090
E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)) ).
Original entry on oeis.org
1, 1, 3, 7, -11, -239, -179, 24991, 192025, -3955391, -89483399, 552615031, 46231717621, 254468241457, -26683006147979, -571848064714289, 14926049610344881, 825004339886219521, -2973711136010539535, -1134313888244827421465, -17734152216328857754739
Offset: 0
-
a(n) = if(n==0, 1, n!*sum(k=1, n, (-n+k+1)^(k-1)*binomial(n-1, n-k)/k!));
A365038
E.g.f. satisfies A(x) = exp(x * (1 + x)/A(x)).
Original entry on oeis.org
1, 1, 1, -2, 9, -44, 175, 246, -21007, 396712, -5576769, 57840850, -151112951, -14137899060, 539212013327, -13335393617714, 239914650459105, -1990873438067504, -76974185162417921, 5220283004540970282, -194958036625254566599, 5226632355735840377140
Offset: 0
Showing 1-6 of 6 results.