A052873 -id:A052873 - cp's OEIS frontend

A365012 E.g.f. satisfies A(x) = exp( xA(x)/(1 - x A(x)^2) ).

1, 1, 5, 52, 833, 18116, 498907, 16648402, 653034545, 29450331928, 1501456530131, 85398143019014, 5361130115439529, 368227694339818132, 27468201247134068891, 2211469648218676671466, 191131823105565504395873, 17650493961604405811144624

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..348

Cf. A052873, A365013.

Array[#!*Sum[ (2 # - k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 19, 0] (* Michael De Vlieger, Aug 18 2023 *)

a(n) = n!*sum(k=0, n, (2*n-k+1)^(k-1)*binomial(n-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (2*n-k+1)^(k-1) * binomial(n-1,n-k)/k!.

A382036 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)^2) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 7, 94, 1901, 51696, 1771267, 73317616, 3560476761, 198531343360, 12502959204671, 877829600807424, 67991178144166213, 5759309535250776064, 529665762441463234875, 52560256640090731902976, 5597859153748148214250673, 636915477940535101583130624, 77102760978489789146276986231

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A052873, A382037, A382038.

Cf. A000108, A377829, A382032.

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(2*n, k)/(n-k-1)!));

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))^2).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^2 ) ).
a(n) ~ 2^(n - 1/2) * n^(n-1) / ((sqrt(2) - 1)^(n - 1/2) * exp((sqrt(2) - 1)*(sqrt(2)*n - 1))). - Vaclav Kotesovec, Mar 15 2025

A382037 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 160, 4325, 157896, 7280077, 406085632, 26599741065, 2001864880000, 170236619802161, 16144762562002944, 1689534516295056301, 193403842876754728960, 24040636567791329323125, 3224829927677539092791296, 464325325579881390473331473, 71428455280041816247241637888

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A052873, A382036, A382038.

Cf. A001764, A377830, A382033.

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(3*n, k)/(n-k-1)!));

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^3).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^3 ) ).

A382038 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^4) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 11, 244, 8285, 381096, 22175167, 1562582848, 129381990201, 12313784396800, 1324663415429651, 158957183013686784, 21051725357219126869, 3050121640032545419264, 479928476696367747954375, 81499293517054315684642816, 14856515462975583258374526833, 2893604521320117995839047401472

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A052873, A382036, A382037.

Cf. A002293, A382034.

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(4*n, k)/(n-k-1)!));

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^4).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^4 ) ).

A364939 E.g.f. satisfies A(x) = exp( xA(x) / (1 - xA(x))^2 ).

Original entry on oeis.org

1, 1, 7, 82, 1421, 32856, 953107, 33316816, 1364109273, 64057409920, 3394727354591, 200445915043584, 13050860745456613, 928976320999078912, 71773343988758253675, 5982029183718123513856, 535011546414154955711153, 51110145581257562326401024

Offset: 0

Seiichi Manyama, Aug 14 2023

nonn

Index entries for reversions of series

Cf. A052873, A364940.

Cf. A082579.

Table[n! * Sum[(n+1)^(k-1) * Binomial[n+k-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 11 2023 *)

a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(n+k-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(n+k-1,n-k)/k!.
a(n) ~ sqrt(((321*(3852 + 215*sqrt(321)))^(1/3) - 321^(2/3)/(3852 + 215*sqrt(321))^(1/3)) / 107) * (4 + ((83 - 3*sqrt(321))/2)^(1/3) + ((83 + 3*sqrt(321))/2)^(1/3))^n * exp(((215 - 12*sqrt(321))^(1/3) + (215 + 12*sqrt(321))^(1/3) - 1) * (n+1)/12 - n) * n^(n-1) / 3^(n + 1/2). - Vaclav Kotesovec, Nov 11 2023
E.g.f.: (1/x) * Series_Reversion( x*exp(-x/(1 - x)^2) ). - Seiichi Manyama, Sep 23 2024

A364940 E.g.f. satisfies A(x) = exp( xA(x) / (1 - xA(x))^3 ).

Original entry on oeis.org

1, 1, 9, 124, 2525, 68616, 2338357, 96004672, 4616135001, 254542038400, 15839013320801, 1098078537291264, 83940831427695541, 7014958697801657344, 636298582947212386125, 62261039244978489081856, 6537251350698278868150833, 733159568772947522820538368

Offset: 0

Seiichi Manyama, Aug 14 2023

nonn

Index entries for reversions of series

Cf. A052873, A364939.

Cf. A091695.

a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(n+2*k-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.

E.g.f.: (1/x) * Series_Reversion( x*exp(-x/(1 - x)^3) ). - Seiichi Manyama, Sep 23 2024

A365013 E.g.f. satisfies A(x) = exp( xA(x)/(1 - x A(x)^3) ).

Original entry on oeis.org

1, 1, 5, 58, 1061, 26536, 843457, 32553424, 1478813513, 77304347776, 4571222616701, 301696674682624, 21985118975444077, 1753288356936334336, 151887264799071753785, 14203597499192539334656, 1426051485043745729079953, 153000280727938469281693696

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..341

Cf. A052873, A365012.

Array[#!*Sum[ (3 # - 2 k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)

a(n) = n!*sum(k=0, n, (3*n-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (3*n-2*k+1)^(k-1) * binomial(n-1,n-k)/k!.

A376474 E.g.f. satisfies A(x) = exp( x^2A(x)^2 / (1 - xA(x)) ).

Original entry on oeis.org

1, 0, 2, 6, 84, 840, 14160, 246960, 5438160, 132209280, 3696265440, 114042297600, 3898083752640, 145315002792960, 5886559994515200, 257081021880883200, 12051082491262214400, 603307920100773888000, 32132914081702520486400, 1814085935013542141952000, 108218538908648830498636800

Offset: 0

Seiichi Manyama, Sep 24 2024

nonn

Index entries for reversions of series

Cf. A052873, A376475.

Cf. A052845.

a(n) = n!*sum(k=0, n\2, (n+1)^(k-1)*binomial(n-k-1, n-2*k)/k!);

E.g.f.: (1/x) * Series_Reversion( x*exp(-x^2 / (1 - x)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (n+1)^(k-1) * binomial(n-k-1,n-2*k)/k!.
a(n) ~ s^2 * (2-r*s) * n^(n-1) / (sqrt(2 - 2*r*s + 4*r^2*s^2 - 4*r^3*s^3 + r^4*s^4) * r^(n-1) * exp(n)), where r = exp(1 - sqrt(7/3) * cos(arctan(3^(-3/2))/3) + sqrt(7) * sin(arctan(3^(-3/2))/3)) * ((1 + sqrt(7) * cos(arctan(3^(3/2))/3) - sqrt(21) * sin(arctan(3^(3/2))/3))/3) = 0.311460490854501594554904428274272083649... and s = exp(-1 + sqrt(7/3) * cos(arctan(3^(-3/2))/3) - sqrt(7) * sin(arctan(3^(-3/2))/3)) = 1.428887069084244135127491236860585605773... - Vaclav Kotesovec, Sep 24 2024

A376475 E.g.f. satisfies A(x) = exp( x^3A(x)^3 / (1 - xA(x)) ).

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 3240, 45360, 584640, 13668480, 322963200, 7224940800, 201040963200, 6254004556800, 197219089267200, 6845849673062400, 260976932536320000, 10410615332941824000, 441056225586706329600, 20015606466369626112000, 955852013167308601344000, 47944066629381635801088000

Offset: 0

Seiichi Manyama, Sep 24 2024

nonn

Index entries for reversions of series

Cf. A052873, A376474.

Cf. A293049.

a(n) = n!*sum(k=0, n\3, (n+1)^(k-1)*binomial(n-2*k-1, n-3*k)/k!);

E.g.f.: (1/x) * Series_Reversion( x*exp(-x^3 / (1 - x)) ).

a(n) = n! * Sum_{k=0..floor(n/3)} (n+1)^(k-1) * binomial(n-2*k-1,n-3*k)/k!.

A376558 E.g.f. satisfies A(x) = exp( xA(x) / (1 - x^2A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 22, 245, 3576, 65527, 1449904, 37596393, 1118442880, 37559084651, 1405597826304, 58012540741597, 2617923512200192, 128240561732097375, 6777245042104293376, 384358793388984148433, 23284761629109883600896, 1500714780345430134323923

Offset: 0

Seiichi Manyama, Sep 28 2024

nonn

Index entries for reversions of series

Cf. A052873, A376563.

Cf. A088009, A376474.

a(n) = n!*sum(k=0, n\2, (n+1)^(n-2*k-1)*binomial(n-k-1, k)/(n-2*k)!);

E.g.f.: (1/x) * Series_Reversion( x*exp(-x / (1 - x^2)) ).

a(n) = n! * Sum_{k=0..floor(n/2)} (n+1)^(n-2*k-1) * binomial(n-k-1,k)/(n-2*k)!.

cp's OEIS Frontend

A365012 E.g.f. satisfies A(x) = exp( x*A(x)/(1 - x * A(x)^2) ).

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A382036 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)^2) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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A382037 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A382038 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^4) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A364939 E.g.f. satisfies A(x) = exp( x*A(x) / (1 - x*A(x))^2 ).

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A364940 E.g.f. satisfies A(x) = exp( x*A(x) / (1 - x*A(x))^3 ).

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A365013 E.g.f. satisfies A(x) = exp( x*A(x)/(1 - x * A(x)^3) ).

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A376474 E.g.f. satisfies A(x) = exp( x^2*A(x)^2 / (1 - x*A(x)) ).

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A376475 E.g.f. satisfies A(x) = exp( x^3*A(x)^3 / (1 - x*A(x)) ).

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A365012 E.g.f. satisfies A(x) = exp( xA(x)/(1 - x A(x)^2) ).

A364939 E.g.f. satisfies A(x) = exp( xA(x) / (1 - xA(x))^2 ).

A364940 E.g.f. satisfies A(x) = exp( xA(x) / (1 - xA(x))^3 ).

A365013 E.g.f. satisfies A(x) = exp( xA(x)/(1 - x A(x)^3) ).

A376474 E.g.f. satisfies A(x) = exp( x^2A(x)^2 / (1 - xA(x)) ).

A376475 E.g.f. satisfies A(x) = exp( x^3A(x)^3 / (1 - xA(x)) ).

A376558 E.g.f. satisfies A(x) = exp( xA(x) / (1 - x^2A(x)^2) ).