A382029
E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
Original entry on oeis.org
1, 1, 3, 31, 529, 12601, 385891, 14440567, 638576065, 32580927505, 1883889232291, 121742057314351, 8695278706372369, 680187946863332233, 57833833258995140803, 5310742450917819399751, 523793286672328763358721, 55223769332070053104438945, 6197871354601209094032190147
Offset: 0
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a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+k, k)/((n+k)*(n-k-1)!)));
A382030
E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.
Original entry on oeis.org
1, 1, 3, 37, 817, 25741, 1053211, 52957297, 3157457185, 217695187801, 17036331544531, 1491702434847901, 144479729938558609, 15335923797225215653, 1770255543485671432555, 220776904683577075549801, 29582947262972619472787521, 4238424613351537181204589745, 646565304924896452410832170787
Offset: 0
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a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+2*k, k)/((n+2*k)*(n-k-1)!)));
A365016
E.g.f. satisfies A(x) = exp( x*A(x)^3/(1 - x * A(x)^2) ).
Original entry on oeis.org
1, 1, 9, 160, 4345, 159796, 7434199, 418864426, 27732988609, 2110729489048, 181587635465671, 17426825999144926, 1845855944285411425, 213900244312057975348, 26919356609721984494311, 3656322063766897691641666, 533110345129065969043548289
Offset: 0
-
Array[#!*Sum[ (2 # + k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 17, 0] (* Michael De Vlieger, Aug 18 2023 *)
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a(n) = n!*sum(k=0, n, (2*n+k+1)^(k-1)*binomial(n-1, n-k)/k!);
A372183
E.g.f. A(x) satisfies A(x) = exp( x * A(x)^5 / (1 - x * A(x)^2) ).
Original entry on oeis.org
1, 1, 13, 340, 13713, 752516, 52372051, 4421017602, 438996446545, 50142716621848, 6477138263806011, 933667525669154486, 148582199464010331289, 25874197258988478298068, 4894174597530612144797299, 999256176035969437218129946, 219035687330062179838536993441
Offset: 0
-
a(n, r=1, s=1, t=5, u=2) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(n+(s-1)*k-1, n-k)/k!);