A361094
E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).
Original entry on oeis.org
1, 1, 9, 166, 4717, 182136, 8911549, 528571408, 36864033945, 2956595372416, 268116203622961, 27128338649300736, 3029974270053623941, 370289278173654092800, 49150116757136815109733, 7041536364582774222616576, 1083004122024520209576760369
Offset: 0
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Table[n! * Sum[(3*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)
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a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(n-1, n-k)/k!);
A361095
E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)) - 1 ).
Original entry on oeis.org
1, 1, 1, -2, -3, 56, -155, -2736, 34489, 72064, -6599799, 53676800, 1155350581, -32238425088, -3604716947, 14790925735936, -235482791871375, -4972572910452736, 254158358486634001, -1028499606209101824, -202204782754527137939, 5371925138905661440000
Offset: 0
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a(n) = if(n==0, 1, n!*sum(k=1, n, (-n+1)^(k-1)*binomial(n-1, n-k)/k!));
A361096
E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)^2) - 1 ).
Original entry on oeis.org
1, 1, -1, 1, 17, -339, 4999, -63587, 566145, 3549241, -405637489, 15518099961, -446235202799, 9617693853925, -75522664207017, -7341781870733099, 596513949276803969, -30104875035438797583, 1144712508931072057375, -27381639204739332379151
Offset: 0
A361097
E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)^3) - 1 ).
Original entry on oeis.org
1, 1, -3, 22, -251, 3816, -71207, 1542640, -36997431, 929097856, -22062115979, 334968255744, 13395424571725, -2177817789105152, 201597999475333329, -16622491076645341184, 1332634806870147259537, -107073894723559010304000
Offset: 0
A363358
E.g.f. satisfies A(x) = exp(x * A(x)^2 * (1 + x * A(x)^2)).
Original entry on oeis.org
1, 1, 7, 91, 1809, 48521, 1643863, 67381875, 3243606817, 179405231761, 11213025902631, 781604862035339, 60120379931640625, 5058593367221610009, 462199816484860893559, 45574025454771003821731, 4823543138131670132557377, 545448517762149418525390625
Offset: 0
A361143
E.g.f. satisfies A(x) = exp( x*A(x)^4/(1 - x*A(x)^2) ).
Original entry on oeis.org
1, 1, 11, 241, 8105, 370061, 21403675, 1500521485, 123685912817, 11724012791929, 1256517775425131, 150254377493878505, 19833528195709809817, 2864566162751107839493, 449364739762263286489403, 76084967168410028438252101, 13829896583435315152843525985
Offset: 0
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
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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!);
A365014
E.g.f. satisfies A(x) = exp( x*A(x)^2/(1 - x * A(x)^3) ).
Original entry on oeis.org
1, 1, 7, 103, 2349, 72961, 2874793, 137399487, 7724650601, 499542475105, 36532938744621, 2981405776356679, 268605245211618637, 26480489709604968129, 2835590837094928349921, 327748240537910056251151, 40669893396736296241364817, 5392699633877586027282801217
Offset: 0
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Array[#!*Sum[ (3 # - k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)
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a(n) = n!*sum(k=0, n, (3*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
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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!);
Showing 1-9 of 9 results.