A354551
Expansion of e.g.f. exp( x * exp(x^3/6) ).
Original entry on oeis.org
1, 1, 1, 1, 5, 21, 61, 211, 1401, 8065, 37241, 240021, 1997821, 13856701, 94418325, 874328911, 8304303281, 69158458881, 658339599601, 7454839614985, 78224066633781, 805961931388741, 9828080719704941, 124199805022959051, 1466207770078872745
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[x*Exp[x^3/6]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 03 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^3/6)))))
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a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*k!*(n-3*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)!));
A356029
a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(2^k * (n - 2*k)!).
Original entry on oeis.org
1, 1, 1, 4, 13, 61, 421, 2626, 27049, 245953, 3069721, 40222216, 576988501, 10058716669, 169773404893, 3596206855606, 73450508303761, 1775382487932001, 43993288886533489, 1183551336464017708, 34806599282992709341, 1043452963148195577181
Offset: 0
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a[n_] := n! * Sum[(n - 2*k)^k/(2^k*(n - 2*k)!), {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)
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a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(2^k*(n-2*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^2/2)))))
A358264
Expansion of e.g.f. 1/(1 - x * exp(x^2/2)).
Original entry on oeis.org
1, 1, 2, 9, 48, 315, 2520, 23415, 248640, 2972025, 39463200, 576413145, 9184855680, 158550787395, 2947473809280, 58707685211175, 1247293022976000, 28156003910859825, 672972205556851200, 16978695795089253225, 450907982644863744000, 12573634144960773960075
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x^2/2))))
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a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(2^k*k!));
A362660
E.g.f. satisfies A(x) = exp( x * exp(x^2/2) * A(x) ).
Original entry on oeis.org
1, 1, 3, 19, 161, 1791, 24847, 413449, 8036625, 178852753, 4486426091, 125279093259, 3854964555697, 129618443364463, 4728625129171959, 186034319795094481, 7851808690935373793, 353903271319498588641, 16966669198377512202643
Offset: 0
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