A354550
Expansion of e.g.f. exp( x * exp(x^2/2) ).
Original entry on oeis.org
1, 1, 1, 4, 13, 46, 241, 1156, 6889, 44668, 300241, 2328976, 18390901, 159273544, 1461200833, 13995753136, 144068872081, 1531949061136, 17259159775969, 202543867724608, 2474236899786781, 31633380519660256, 417760492214548561, 5751414293905728064
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[x Exp[x^2/2]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 03 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^2/2)))))
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a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(2^k*k!*(n-2*k)!));
A354552
Expansion of e.g.f. exp( x * exp(x^4/24) ).
Original entry on oeis.org
1, 1, 1, 1, 1, 6, 31, 106, 281, 946, 7561, 54286, 281161, 1207636, 7997991, 81996916, 701522641, 4580581916, 29742355441, 306369616636, 3632198902321, 34710574441096, 276645112305871, 2652825718776696, 35647605796451881, 458142859493786776
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^4/24)))))
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a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(24^k*k!*(n-4*k)!));
A356328
a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(6^k * (n - 3*k)!).
Original entry on oeis.org
1, 1, 1, 1, 5, 21, 61, 281, 2521, 15625, 84841, 971521, 10646461, 83366141, 962405445, 15445935961, 181502928881, 2182235585041, 42297481449361, 714940186390465, 10007476059187381, 204722588272279141, 4600003555996715021, 80767827313930590681
Offset: 0
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a[n_] := n! * Sum[(n - 3*k)^k/(6^k*(n - 3*k)!), {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 24, 0] (* Amiram Eldar, Aug 19 2022 *)
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a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*(n-3*k)!));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^3/6)))))
A358265
Expansion of e.g.f. 1/(1 - x * exp(x^3/6)).
Original entry on oeis.org
1, 1, 2, 6, 28, 160, 1080, 8470, 76160, 771120, 8671600, 107245600, 1446984000, 21150929800, 332950217600, 5615507898000, 101024594070400, 1931055071545600, 39082823446867200, 834945681049480000, 18776164188349568000, 443348081412556320000
Offset: 0
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g := 1/(1-x*exp(x^3/6)) ;
taylor(%,x=0,70) ;
L := gfun[seriestolist](%) ;
seq( op(i,L)*(i-1)!,i=1..nops(L)) ; # R. J. Mathar, Mar 13 2023
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x^3/6))))
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a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*k!));
A362661
E.g.f. satisfies A(x) = exp( x * exp(x^3/6) * A(x) ).
Original entry on oeis.org
1, 1, 3, 16, 129, 1356, 17767, 279714, 5149209, 108591688, 2582351451, 68380940904, 1995777685717, 63659665732716, 2203395556479951, 82253291389678756, 3294326092613575473, 140911264444599281616, 6411278790217738946899
Offset: 0
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