A351929
Expansion of e.g.f. exp(x - x^3/6).
Original entry on oeis.org
1, 1, 1, 0, -3, -9, -9, 36, 225, 477, -819, -10944, -37179, 16875, 870507, 4253796, 2481921, -101978919, -680495175, -1060229088, 16378166061, 145672249311, 368320357791, -3415036002300, -40270115077983, -141926533828299, 882584266861701, 13970371667206176
Offset: 0
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m = 27; Range[0, m]! * CoefficientList[Series[Exp[x - x^3/6], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^3/6)))
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a(n) = n!*sum(k=0, n\3, (-1/3!)^k*binomial(n-2*k, k)/(n-2*k)!);
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a(n) = if(n<3, 1, a(n-1)-binomial(n-1, 2)*a(n-3));
A351931
Expansion of e.g.f. exp(x - x^5/120).
Original entry on oeis.org
1, 1, 1, 1, 1, 0, -5, -20, -55, -125, -125, 925, 7525, 34750, 124125, 249250, -1013375, -14708875, -97413875, -477236375, -1443329375, 3466472500, 91499089375, 804081585000, 5030009685625, 20366827624375, -23484049500625, -1391395435656875, -15503027252406875
Offset: 0
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m = 28; Range[0, m]! * CoefficientList[Series[Exp[x - x^5/5!], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^5/5!)))
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a(n) = n!*sum(k=0, n\5, (-1/5!)^k*binomial(n-4*k, k)/(n-4*k)!);
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a(n) = if(n<5, 1, a(n-1)-binomial(n-1, 4)*a(n-5));
A351932
Number of set partitions of [n] such that block sizes are either 1 or 4.
Original entry on oeis.org
1, 1, 1, 1, 2, 6, 16, 36, 106, 442, 1786, 6106, 23596, 120836, 631632, 2854216, 13590396, 81258556, 510768316, 2839808572, 16008902296, 108643656136, 787965516416, 5161270717296, 33513036683512, 253407796702776, 2065728484459576, 15485032349429176, 113510664648701776
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 1,
`if`(n<4, 0, a(n-4)*binomial(n-1, 3))+a(n-1))
end:
seq(a(n), n=0..28); # Alois P. Heinz, Feb 26 2022
seq(round(evalf(hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3))),n=0..28); # Karol A. Penson, Jul 28 2023
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x+x^4/4!)))
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a(n) = n!*sum(k=0, n\4, 1/4!^k*binomial(n-3*k, k)/(n-3*k)!);
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a(n) = if(n<4, 1, a(n-1)+binomial(n-1, 3)*a(n-4));
A362342
a(n) = n! * Sum_{k=0..floor(n/4)} (-k/24)^k / (k! * (n-4*k)!).
Original entry on oeis.org
1, 1, 1, 1, 0, -4, -14, -34, 71, 1135, 6091, 22771, -87119, -1847559, -13769755, -70046339, 390688481, 10473961121, 100030347361, 643972996705, -4717305354419, -153449916040259, -1787926183752939, -13926752488607419, 126329848106764765
Offset: 0
A362345
a(n) = n! * Sum_{k=0..floor(n/4)} (-n/24)^k /(k! * (n-4*k)!).
Original entry on oeis.org
1, 1, 1, 1, -3, -24, -89, -244, 1681, 24382, 155401, 695146, -7490339, -157336464, -1421454033, -8817579224, 129268310081, 3555528110716, 41578411339441, 329824291072252, -6116622750516899, -207991913454970784, -2985298421745508329
Offset: 0
Showing 1-5 of 5 results.