A082579 -id:A082579 - cp's OEIS frontend

A361598 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

1, 2, 9, 58, 473, 4626, 52537, 677594, 9762993, 155175778, 2693718281, 50657791482, 1025158123849, 22198908725618, 511885585833273, 12517101011344666, 323402336324055137, 8800318580852865474, 251497162228635927433, 7529081846683064675258

Offset: 0

Seiichi Manyama, Mar 17 2023

nonn

Column k=2 of A361600.

Cf. A082579.

Table[n! * Sum[Binomial[n+k,2*k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^2)/(1-x)))

a(n) = n! * Sum_{k=0..n} binomial(n+k,2*k)/k! = Sum_{k=0..n} (n+k)!/(2*k)! * binomial(n,k).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 1)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/6) * (1 + 1/(2^(2/3)*n^(1/3)) + 83/(360*2^(1/3)*n^(2/3))). (End)

A362775 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x) ).

Original entry on oeis.org

1, 1, 7, 70, 965, 17216, 379207, 9969772, 305154313, 10668593008, 419714689931, 18358646058644, 884070662867053, 46486344447041032, 2650567497877525423, 162908800485532424236, 10737607698626311094033, 755571950776792829919968

Offset: 0

Seiichi Manyama, May 02 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A082579, A362776.

Cf. A052868, A361065.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^2))))

E.g.f.: exp( -LambertW(-x/(1-x)^2) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k-1,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: -LambertW(-x/(1-x)^2) * (1-x)^2 / x.
a(n) ~ 2^(n + 1/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n - 1/2) * exp(2*n-1)). (End)

A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 481, 3841, 38221, 464353, 5368609, 82042561, 1151767981, 20242097041, 342921513793, 6705416722369, 133590317946541, 2880298682358721, 65597610230669761, 1556262483879791233, 39569880403136366029, 1030778206965403668721

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A012150, A088009, A373619.

Cf. A082579, A373578.

A373620 := proc(n)
    add(binomial(2*n-3*k-1,k)/(n-2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A373620(n),n=0..80) ; # R. J. Mathar, Jun 11 2024

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

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k)/(n-2*k)!.
a(n) == 1 mod 12.
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/48 + 2^(-5/3)*n^(1/3) + 3*2^(-4/3)*n^(2/3) - n) * n^(n - 1/6). - Vaclav Kotesovec, Jun 11 2024
D-finite with recurrence a(n) -a(n-1) -3*(n-1)*(n-2)*a(n-2) -3*(n-1)*(n-2)*a(n-3) +3*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jun 11 2024

A380511 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 55, 961, 23141, 711421, 26631235, 1175535425, 59786520841, 3442729157461, 221413508687471, 15730688410899265, 1223574846548300845, 103417508018836074701, 9437941200860641295611, 924934291227615821904001, 96881241931552168636182545, 10801002623361396194857667365

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A251569, A380512.
Cf. A080893, A380515.
Cf. A082579, A251568, A380514.
Cf. A001764, A006013, A370054, A382058.

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

a(n) = 2 * n! * Sum_{k=0..n-1} binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-3*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^2 ) ). - Seiichi Manyama, Mar 15 2025

A353162 Expansion of e.g.f. exp(Sum_{p prime} p * x^p).

Original entry on oeis.org

1, 0, 4, 18, 48, 1320, 4200, 115920, 994560, 11793600, 264055680, 2601244800, 67761429120, 1067726499840, 21513457405440, 485310649824000, 9925206939648000, 254012624170905600, 6174538264806912000, 160933619800835481600, 4458470291543671603200

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A082579, A218002, A351940, A353163, A353164.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*k*x^k))))

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

a(0) = 1; a(n) = (n-1)! * Sum_{p<=n, p prime} p^2 * a(n-p)/(n-p)!.

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

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

Original entry on oeis.org

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800

Offset: 0

Vaclav Kotesovec, Aug 24 2025

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293012, A000262, A082579, A091695, A361283.

Cf. A052845, A052887, A386514.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025

nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

A270669 E.g.f.: Product_{k>=1} (1 + sinh(k*x^k)).

Original entry on oeis.org

1, 1, 4, 31, 168, 1841, 19320, 226885, 2655408, 47569681, 743996880, 12582916061, 245804712120, 4831304993113, 109782586920552, 2669560767444901, 61579705719702240, 1566459883903878305, 44585240861695115808, 1212424119941953292461, 37517727808419084095400

Offset: 0

Vaclav Kotesovec, Mar 21 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Cf. A082579, A130220, A130263, A255807, A270294, A270670.

nn = 25; Range[0, nn]! * CoefficientList[Series[Product[1+Sinh[k*x^k], {k, 1, nn}], {x, 0, nn}], x]

A293785 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j>=1} j^(k-1)*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 5, 13, 24, 1, 1, 9, 31, 73, 120, 1, 1, 17, 79, 241, 501, 720, 1, 1, 33, 211, 841, 2261, 4051, 5040, 1, 1, 65, 583, 3049, 10821, 24781, 37633, 40320, 1, 1, 129, 1651, 11353, 54221, 162601, 309835, 394353, 362880, 1, 1, 257

Offset: 0

Seiichi Manyama, Oct 16 2017

			Square array begins:
     1,   1,    1,     1,     1, ...
     1,   1,    1,     1,     1, ...
     2,   3,    5,     9,    17, ...
     6,  13,   31,    79,   211, ...
    24,  73,  241,   841,  3049, ...
   120, 501, 2261, 10821, 54221, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000142, A000262, A082579, A255807, A255819.
Rows n=0-1 give A000012.
Main diagonal gives A293786.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j^k*A(n-j,k)/(n-j)! for n > 0.

A328054 Expansion of e.g.f. log(1 + x / (1 - x)^2).

Original entry on oeis.org

0, 1, 3, 8, 18, 24, 0, 720, 15120, 161280, 1088640, 3628800, 0, 479001600, 18681062400, 348713164800, 3923023104000, 20922789888000, 0, 6402373705728000, 364935301226496000, 9731608032706560000, 153272826515128320000, 1124000727777607680000, 0, 620448401733239439360000

Offset: 0

Ilya Gutkovskiy, Oct 03 2019

nonn

Logarithmic transform of A001563.

Robert Israel, Table of n, a(n) for n = 0..450

Cf. A001563, A008588 (positions of 0's), A009306, A082579, A328055.

b:= proc(n) option remember; n*n! end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-
      add(binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 04 2019

nmax = 25; CoefficientList[Series[Log[1 + x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[2 (n - 1)! (1 - Cos[Pi n/3]), {n, 1, 25}]]

my(x='x+O('x^30)); concat(0, Vec(serlaplace(log(1 + x / (1 - x)^2)))) \\ Michel Marcus, Oct 04 2019

E.g.f.: log(1 + Sum_{k>=1} k * x^k).

D-finite with recurrence a(n+3) = n*(n+1)*(n+2)*a(n) - 2*(n+2)*(n+1)*a(n+1) + 2*(n+2)*a(n+2). - Robert Israel, Jan 16 2023

cp's OEIS Frontend

A361598 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

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A362775 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x) ).

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A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

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Maple

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A380511 Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A353162 Expansion of e.g.f. exp(Sum_{p prime} p * x^p).

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

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A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

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Magma

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A270669 E.g.f.: Product_{k>=1} (1 + sinh(k*x^k)).

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A293785 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j>=1} j^(k-1)*x^j).

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A380511 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

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