A052845 -id:A052845 - cp's OEIS frontend

A361532 Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

1, 1, 4, 19, 118, 886, 7786, 78184, 881644, 11017108, 150966856, 2249261356, 36181351504, 624658612384, 11516406883528, 225740649754936, 4686671645814736, 102712289940757264, 2369128149877075264, 57359541280704038128, 1454229915957292684576

Offset: 0

Seiichi Manyama, Mar 15 2023

nonn

Cf. A361548, A361557, A361558.

Cf. A052845, A112242, A185369.

With[{nn=20},CoefficientList[Series[Exp[(x+x^2/2)/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 08 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x+x^2/2)/(1-x))))

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - binomial(n-1,2) * a(n-3) for n > 2.

a(n) ~ 2^(-3/4) * 3^(1/4) * exp(-5/4 + sqrt(6*n) - n) * n^(n - 1/4) * (1 + sqrt(3)/(2*sqrt(2*n))). - Vaclav Kotesovec, Mar 20 2023

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

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).

A113237 E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

Original entry on oeis.org

1, 1, 3, 13, 49, 381, 2971, 26713, 291873, 3262969, 41245651, 569262981, 8433896593, 136060620853, 2342471665899, 42987065380561, 838321137046081, 17272648375895793, 375413770580941603, 8579701021461918589, 205637099039964274161, 5158188565847339152621

Offset: 0

Karol A. Penson, Oct 19 2005

nonn

Number of partitions of {1,..,n} into any number of lists of size not equal to 4, where a list means an ordered subset, cf. A000262.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A052845, A097514, A113235, A113236.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!* CoefficientList[ Series[ Exp[x*(1 - x^3 + x^4)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}], n=0, 1....
Recurrence: a(n) = (2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) - 4*(n-3)*(n-2)*(n-1)*a(n-4) + 8*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Jun 24 2013
a(n) ~ n^(n-1/4)*exp(-3/2+2*sqrt(n)-n)/sqrt(2) * (1 + 187/(48*sqrt(n))). - Vaclav Kotesovec, Jun 24 2013

A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 36, 24, 12, 0, 1, 240, 180, 60, 20, 0, 1, 1920, 1440, 540, 120, 30, 0, 1, 17640, 13440, 5040, 1260, 210, 42, 0, 1, 183120, 141120, 53760, 13440, 2520, 336, 56, 0, 1, 2116800, 1648080, 635040, 161280, 30240, 4536, 504, 72, 0, 1

Offset: 0

Vladeta Jovovic, Feb 06 2006

The average number of size 1 lists goes to 1 as n->infinity. In other words, lim_{n->infinity} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. - Geoffrey Critzer, Feb 20 2022 (after asymptotic limits by Vaclav Kotesovec given in A000262)

			Triangle begins:
    1;
    0,   1;
    2,   0,  1;
    6,   6,  0,  1;
   36,  24, 12,  0, 1;
  240, 180, 60, 20, 0, 1;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 0..5150

Cf. A000262, A006152, A052845, A052852.

t:=taylor(exp(x/(1-x)+(y-1)*x),x,11):for n from 0 to 10 do for k from 0 to n do printf("%d, ",coeff(n!*coeff(t,x,n),y,k)): od: printf("\n"): od: # Nathaniel Johnston, Apr 27 2011
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=1, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

nn = 10; Table[Take[(Range[0, nn]! CoefficientList[ Series[Exp[ x/(1 - x) - x + y x], {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}] // Grid (* Geoffrey Critzer, Feb 19 2022 *)

E.g.f.: exp(x/(1-x)+(y-1)*x). More generally, e.g.f. for number of partitions of n-set into lists with k lists of size m is exp(x/(1-x)+(y-1)*x^m).

A376494 E.g.f. satisfies A(x) = exp(x^2 * A(x)^2 / (1 - x)).

Original entry on oeis.org

1, 0, 2, 6, 84, 720, 12000, 178920, 3744720, 79531200, 2056652640, 56284351200, 1753673423040, 58443081016320, 2142625074670080, 83948606126985600, 3549356731374854400, 159643527455123712000, 7656564912324122995200

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

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

Cf. A052868, A376495.

Cf. A052845, A376474.

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

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

E.g.f.: exp( -LambertW(-2*x^2 / (1-x))/2 ).
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * binomial(n-k-1,n-2*k)/k!.
a(n) ~ sqrt(16 + 2*exp(-1) - 2*exp(-1/2)*sqrt(exp(-1)+8)) * (exp(1/2)*sqrt(exp(-1)+8) - 1) * 2^(2*n-2) * n^(n-1) / ((4 + exp(-1) - exp(-1/2)*sqrt(exp(-1)+8)) * (sqrt(1 + 8*exp(1)) - 1)^n). - Vaclav Kotesovec, Aug 05 2025

A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

Original entry on oeis.org

1, 0, 2, 18, 156, 1560, 18480, 254520, 3973200, 68947200, 1312748640, 27175024800, 607314818880, 14566195163520, 373027570755840, 10154293067318400, 292659790712889600, 8899747730037964800, 284685195814757337600, 9553060139009702515200, 335468448755976164428800

Offset: 0

Enrique Navarrete, Aug 23 2025

For n > 0, a(n) is the number of ways to linearly order n distinguishable objects into one or several lines and then choose 2 objects from each line. If the lines are also linearly ordered see A364524.

A001804(n) is the number of ways if only 1 line is used.

			a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.

Cf. A001804, A364524.
Cf. A293012, A000262, A082579, A091695, A361283.
Cf. A052845, A052887, A387244.

nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)

From Vaclav Kotesovec, Aug 24 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 3*k-1) / k!.
a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-7)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(1/27 - 3^(-5/4)*n^(1/4)/8 - 3^(-1/2)*n^(1/2)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n-1/8) / 2. (End)

A076126 Triangle T(n,k) of associated Lah numbers, n>=2, k=1..floor(n/2).

Original entry on oeis.org

2, 6, 24, 12, 120, 120, 720, 1080, 120, 5040, 10080, 2520, 40320, 100800, 40320, 1680, 362880, 1088640, 604800, 60480, 3628800, 12700800, 9072000, 1512000, 30240, 39916800, 159667200, 139708800, 33264000, 1663200, 479001600

Offset: 2

Vladeta Jovovic, Vladimir Baltic, Oct 31 2002

Number of partitions of {1,..,n} into k lists of size >1, where a list means an ordered subset, cf. A008297.

			2; 6; 24, 12; 120,120; 720,1080,120; 5040,10080, 2520; ...

Row sums give A052845, A008306, A008299.

T(n, k) = n!/k!*binomial(n-k-1, k-1), n>=2, k=1..floor(n/2). G.f.: G.f.: Sum_{n>=2, k=1..floor(n/2)} T(n, k)*x^n*y^k/n! = exp(x^2*y/(1-x))-1.

A271706 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*L(j, k), L the unsigned Lah numbers A271703, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -1, 3, 3, 1, 1, 8, 18, 8, 1, -1, 45, 110, 70, 15, 1, 1, 264, 795, 640, 195, 24, 1, -1, 1855, 6489, 6335, 2485, 441, 35, 1, 1, 14832, 59332, 67984, 32550, 7504, 868, 48, 1, -1, 133497, 600732, 789852, 445914, 126126, 19068, 1548, 63, 1

Offset: 0

Peter Luschny, Apr 20 2016

			Triangle starts:
  [ 1]
  [-1,    1]
  [ 1,    0,    1]
  [-1,    3,    3,    1]
  [ 1,    8,   18,    8,    1]
  [-1,   45,  110,   70,   15,   1]
  [ 1,  264,  795,  640,  195,  24,  1]
  [-1, 1855, 6489, 6335, 2485, 441, 35, 1]

A052845 (row sums), A000240 (col. 1), A000274 (col. 2), A067998 (diag n,n-1).

Cf. A271703, A271705.

L := (n, k) -> `if`(k<0 or k>n, 0, (n-k)!*binomial(n, n-k)*binomial(n-1, n-k)):
T := (n, k) -> add(L(j, k)*binomial(-j-1,-n-1), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);
# Or:
T := (n, k) -> (-1)^(n-k)*binomial(n, k)*hypergeom([k-n, k], [], 1):
for n from 0 to 8 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Jun 25 2025

T(n, k) = (-1)^(k-n)*binomial(n, k)*hypergeom([k-n, k], [], 1). (After a formula of Natalia L. Skirrow in A271705.) - Peter Luschny, Jun 25 2025

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A361532 Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

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

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A113237 E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

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A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

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A376494 E.g.f. satisfies A(x) = exp(x^2 * A(x)^2 / (1 - x)).

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A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

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A076126 Triangle T(n,k) of associated Lah numbers, n>=2, k=1..floor(n/2).

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