A052859 -id:A052859 - cp's OEIS frontend

A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).

1, 0, 1, 3, 10, 45, 241, 1428, 9325, 67035, 524926, 4429953, 40010785, 384853560, 3925008361, 42270555603, 478998800290, 5693742545445, 70804642315921, 918928774274028, 12419848913448565, 174467677050577515, 2542777209440690806, 38388037137038323353

Offset: 0

Vladeta Jovovic, Mar 27 2001

nonn

After the first term, this is the Stirling transform of the sequence of moments of the standard normal (or "Gaussian") probability distribution. It is not itself a moment sequence of any probability distribution. - Michael Hardy (hardy(AT)math.umn.edu), May 29 2005

a(n) is the number of simple labeled graphs on n nodes in which each component is a complete bipartite graph. - Geoffrey Critzer, Dec 03 2011

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, Ex. 3.3.5b.

Alois P. Heinz, Table of n, a(n) for n = 0..518 (first 101 terms from Harry J. Smith)
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A324162.

Cf. A052859, A330047.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 2), j=2..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2019

a = Exp[x] - 1; Range[0, 20]! CoefficientList[Series[Exp[a^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Dec 03 2011 *)

a(n)=if(n<0, 0, n!*polcoeff( exp((exp(x+x*O(x^n))-1)^2/2), n)) /* Michael Somos, Jun 01 2005 */

{ for (n=0, 100, write("b060311.txt", n, " ", n!*polcoeff(exp((exp(x + x*O(x^n)) - 1)^2/2), n)); ) } \\ Harry J. Smith, Jul 03 2009

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/(2^k*k!)); \\ Seiichi Manyama, May 07 2022

E.g.f. A(x) = B(exp(x)-1) where B(x)=exp(x^2/2) is e.g.f. of A001147(2n), hence a(n) is the Stirling transform of A001147(2n). - Michael Somos, Jun 01 2005
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp(1/2*(exp(r)-1)^2 - n) * n^(n+1/2) / (r^n * sqrt(exp(r)*r*(-1-r+exp(r)*(1+2*r)))), where r is the root of the equation exp(r)*(exp(r) - 1)*r = n.
(a(n)/n!)^(1/n) ~ 2*exp(1/LambertW(2*n)) / LambertW(2*n).
(End)
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/(2^k * k!). - Seiichi Manyama, May 07 2022

A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 900, 9366, 101556, 1031190, 10995300, 134640726, 1844184276, 26656678230, 400614423300, 6347263038486, 106960986110196, 1905688502565270, 35546025523227300, 691014283378745046, 13999772792477879316, 295570215436360196310

Offset: 0

Seiichi Manyama, May 07 2022

nonn

Cf. A000110, A052859, A353665.

Cf. A327504, A353344, A353774.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^3)))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(k!*prod(j=1, 3*k, 1-j*x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 2)*v[i-j+1])); v;

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

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(k! * Product_{j=1..3*k} (1 - j * x)).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/k!.

A353665 Expansion of e.g.f. exp((exp(x) - 1)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 1560, 8400, 60984, 912240, 15938520, 242998800, 3300493944, 44583979440, 690641504280, 12868117189200, 264164524958904, 5481631005177840, 112822632387018840, 2367468210865875600, 52624238539033647864, 1258531092544541563440

Offset: 0

Seiichi Manyama, May 07 2022

nonn

Cf. A000110, A052859, A353664.

Cf. A327505, A353358, A353775.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^4)))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(k!*prod(j=1, 4*k, 1-j*x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 4, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/k!);

E.g.f.: exp((exp(x) - 1)^4).
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(k! * Product_{j=1..4*k} (1 - j * x)).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,4) * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/k!.

A357024 E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^2.

Original entry on oeis.org

1, 0, 2, 6, 74, 750, 11402, 195006, 3994202, 93164910, 2455754762, 72098755806, 2333497474970, 82569245246670, 3170700672801482, 131342693516044926, 5837883571730770778, 277151780512413426990, 13997018265350140886282, 749304617892345721184286

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A030019, A357025.

Cf. A052859, A357009, A357031.

m = 20; (* number of terms *)
A[_] = 0;
Do[A[x_] = Exp[(Exp[x*A[x]] - 1)^2] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

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

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n+1)^(k-1) * Stirling2(n,2*k)/k!.

A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! Stirling2(n,k*j)/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0

Offset: 0

Seiichi Manyama, Oct 17 2022

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  1,   0,   0,   0,   0, ...
  0,  2,   2,   0,   0,   0, ...
  0,  5,   6,   6,   0,   0, ...
  0, 15,  26,  36,  24,   0, ...
  0, 52, 150, 150, 240, 120, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns k=0-4 give: A000007, A000110, A052859, A353664, A353665.

Cf. A324162, A357293, A357868.

T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);

T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));

For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).

A086660 Stirling transform of Hermite numbers: Sum_{k=0..n} Stirling2(n,k) * HermiteH(k,0).

Original entry on oeis.org

1, 0, -2, -6, -2, 90, 598, 1554, -10082, -164310, -1101242, -1496286, 64767118, 965876730, 7104497398, 57428274, -856472198402, -14195316779190, -122409183339482, 25272908324034, 21770258523698158

Offset: 0

Vladeta Jovovic, Sep 12 2003

sign

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A067994, A052859.

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-(Exp(x)-1)^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 12 2018

Table[Sum[StirlingS2[n,k]HermiteH[k,0],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 24 2013 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-(Exp[x] - 1)^2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 12 2018 *)

x='x+O('x^50); Vec(serlaplace(exp(-(exp(x)-1)^2))) \\ G. C. Greubel, Jul 12 2018

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

A357009 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^2 * A(x).

Original entry on oeis.org

1, 0, 2, 6, 50, 390, 4322, 53046, 782210, 12920550, 241747682, 5000171286, 113961184130, 2830240421190, 76196913418082, 2209152734071926, 68655746019566210, 2276606079902438310, 80244521295497399522, 2995966456305973559766, 118119901491333724203650

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

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

Cf. A052880, A357010.

Cf. A052859, A357024.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])^2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(2*k)/k!)))

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

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

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (k+1)^(k-1) * Stirling2(n,2*k)/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(2*k) / k!.
E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^2) ).
E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^2)/(exp(x) - 1)^2.
a(n) ~ sqrt(1 + exp(1/2)) * 2^n * n^(n-1) / (exp(n-1) * (2*log(1 + exp(1/2)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Sep 27 2023

A375773 Expansion of e.g.f. exp((exp(x) - 1)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 6917400, 129399600, 3259080000, 72252300120, 1370602233000, 23218349918400, 377834084082000, 6709735404918120, 147369456297228600, 3899127761438053200, 109421543771265852000, 3002806840023201408120

Offset: 0

Seiichi Manyama, Aug 27 2024

nonn

Cf. A000110, A052859, A353664, A353665.

Cf. A353404, A373940.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^5)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/k!);

G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(k! * Product_{j=1..5*k} (1 - j * x)).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/k!.

cp's OEIS Frontend

A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).

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A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

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A353665 Expansion of e.g.f. exp((exp(x) - 1)^4).

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A357024 E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^2.

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A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.

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A086660 Stirling transform of Hermite numbers: Sum_{k=0..n} Stirling2(n,k) * HermiteH(k,0).

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A357009 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^2 * A(x).

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A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! Stirling2(n,k*j)/j!.