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Expansion of e.g.f. 1/(1 - x * exp(x^2)).
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11
1, 1, 2, 12, 72, 540, 5040, 53760, 658560, 9087120, 139104000, 2343781440, 43078210560, 857676980160, 18390744852480, 422504399116800, 10353592759910400, 269576216304595200, 7431814422621388800, 216266552026593868800, 6624610236968435712000
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/k!.
a(n) ~ n! * 2^(n/2) / ((1 + LambertW(2)) * LambertW(2)^(n/2)). - Vaclav Kotesovec, Nov 01 2022
MATHEMATICA
With[{nn=30}, CoefficientList[Series[1/(1-x Exp[x^2]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Feb 14 2024 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x^2))))
(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k/k!);
a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(n - 3*k)!.
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6
1, 1, 1, 1, 25, 121, 361, 5881, 82321, 547345, 6053041, 167991121, 2179469161, 22892967241, 788375451865, 18046198202761, 245523704069281, 7548055281543841, 270833271588545761, 5369819950838359585, 141456920470310708281, 6760255576117937586841
FORMULA
E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^3)).
a(n) ~ sqrt(Pi/3) * exp((2*n - 3)/(6*LambertW(exp(1/4)*(2*n - 3)/8)) - 4*n/3) * n^(4*n/3 + 1/2) / (sqrt(1 + LambertW(exp(1/4)*(2*n - 3)/8)) * 2^(2*n/3 + 1/2) * LambertW(exp(1/4)*(2*n - 3)/8)^(n/3)). - Vaclav Kotesovec, Nov 01 2022
MATHEMATICA
a[n_] := n! * Sum[(n - 3*k)^k/(n - 3*k)!, {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(n-3*k)!);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^3)))))
E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^3*A(x)^3).
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6
1, 1, 2, 6, 48, 720, 11520, 183960, 3185280, 65681280, 1637193600, 46436544000, 1423113753600, 46607434473600, 1648149184281600, 63369409495392000, 2634451417524326400, 117088187211284889600, 5518546983426135859200, 275022667579200532992000
FORMULA
a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n+1,n-3*k)/k!.
a(n) ~ 3^(n/3) * (1 + 4*LambertW(3^(1/4)/4))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(3^(1/4)/4)) * 2^(8*n/3 + 4) * exp(n) * LambertW(3^(1/4)/4)^(4*n/3 + 3/2)). - Vaclav Kotesovec, Nov 08 2023
MATHEMATICA
Join[{1}, Table[n!/(n+1) * Sum[(n-3*k)^k * Binomial[n+1, n-3*k]/k!, {k, 0, Floor[n/3]}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 08 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n+1, n-3*k)/k!)/(n+1);
Expansion of e.g.f. 1/(1 - x^3 * exp(x)).
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1, 0, 0, 6, 24, 60, 840, 10290, 80976, 847224, 13306320, 190271070, 2677088040, 46082426676, 874515884424, 16582066303530, 336875275380000, 7539189088358640, 176554878235711776, 4295134487197296054, 111114287924643309240, 3036073975138066955820
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/(n - 3*k)!.
a(n) ~ n! / ((1 + LambertW(1/3)) * 3^(n+1) * LambertW(1/3)^n). - Vaclav Kotesovec, Oct 30 2022
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*exp(x))))
(PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(n-3*k)!);
E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^3*A(x)).
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4
1, 1, 2, 6, 48, 480, 5040, 57960, 806400, 13426560, 250992000, 5102697600, 113283878400, 2760905347200, 73287883468800, 2093750122464000, 63947194517606400, 2082970788291993600, 72182922107859763200, 2651026034089585152000
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n-2*k+1,n-3*k)/( (n-2*k+1)*k! ).
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n-2*k+1, n-3*k)/((n-2*k+1)*k!));
Expansion of e.g.f. 1/(1 - x * exp(x^3/6)).
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3
1, 1, 2, 6, 28, 160, 1080, 8470, 76160, 771120, 8671600, 107245600, 1446984000, 21150929800, 332950217600, 5615507898000, 101024594070400, 1931055071545600, 39082823446867200, 834945681049480000, 18776164188349568000, 443348081412556320000
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(6^k * k!).
a(n) ~ n! / ((1 + LambertW(1/2)) * (2*LambertW(1/2))^(n/3)). - Vaclav Kotesovec, Nov 13 2022
MAPLE
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
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x^3/6))))
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*k!));
E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^3*A(x)^2).
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3
1, 1, 2, 6, 48, 600, 7920, 108360, 1693440, 32114880, 715478400, 17616614400, 467505561600, 13438170345600, 421361740800000, 14345678194848000, 524464774215782400, 20420391682852761600, 844038690729589555200, 36981569420732192256000
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n-k+1,n-3*k)/( (n-k+1)*k! ).
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n-k+1, n-3*k)/((n-k+1)*k!));
Expansion of e.g.f. exp(2*x^3) / (1 - x * exp(x^3)).
+10
1
1, 1, 2, 18, 96, 600, 5760, 57960, 645120, 8285760, 117936000, 1842825600, 31374604800, 578334556800, 11493004723200, 244720360684800, 5555523785011200, 134002274473267200, 3422904611167641600, 92290617116425728000, 2619214995575033856000
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+2)^k/k!.
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(2*x^3)/(1-x*exp(x^3))))
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k+2)^k/k!);
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