A052880 -id:A052880 - cp's OEIS frontend

A033917 Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1.

1, 1, 2, 9, 56, 480, 5094, 65534, 984808, 16992144, 330667680, 7170714672, 171438170232, 4480972742064, 127115833240200, 3889913061111240, 127729720697035584, 4479821940873927168, 167143865005981109952, 6610411989494027218368, 276242547290322145178880

Offset: 0

Patrick D McLean

a(n) is the n-th derivative of x^(x^...(x^(x^x))) with n x's evaluated at x=1. - Alois P. Heinz, Oct 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..400
R. Arthur Knoebel, Exponentials Reiterated, Amer. Math. Monthly 88 (1981), pp. 235-252.
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Cf. A052880, A215703, A216349, A216350.
Row sums of A277536.
Main diagonal of A277537.

a:= n-> add(Stirling1(n, k)*(k+1)^(k-1), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 31 2012

mx = 20; Table[ i! SeriesCoefficient[ InverseSeries[ Series[ y^(1/y), {y, 1, mx}]], i], {i, 0, n}] (* modified by Robert G. Wilson v, Feb 03 2013 *)
CoefficientList[Series[-LambertW[-Log[1+x]]/Log[1+x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)
a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Mar 09 2013

E.g.f.: -LambertW(-log(1+x))/log(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ n^(n-1) / ( exp(n -3/2 + exp(-1)/2) * (exp(exp(-1))-1)^(n-1/2) ). - Vaclav Kotesovec, Nov 27 2012
E.g.f.: A(x) satisfies A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling1(n,k) * A(x)^k. - Paul D. Hanna, Mar 09 2013
a(n) = n! * [x^n] (x+1)^^n. - Alois P. Heinz, Oct 19 2016

A135313 Triangle of numbers T(n,k) (n>=0, n>=k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where k=max_{x}(|{y : xRy}|), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 12, 13, 0, 1, 61, 106, 75, 0, 1, 310, 1105, 1035, 541, 0, 1, 1821, 12075, 16025, 11301, 4683, 0, 1, 11592, 141533, 267715, 239379, 137774, 47293, 0, 1, 80963, 1812216, 4798983, 5287506, 3794378, 1863044, 545835, 0, 1, 608832, 25188019, 92374107, 124878033, 105494886, 64432638, 27733869, 7087261

Offset: 0

Alois P. Heinz, Dec 05 2007

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Conjecture: For fixed k>=0, A135313(n+k,n) ~ n! * n^(2*k) / (2^(k+1) * k! * log(2)^(n+k+1)). - Vaclav Kotesovec, Nov 20 2021

			T(3,3) = 13 because there are 13 relations of the given kind for 3 elements:  (1) 1R2, 2R1, 1R3, 3R1, 2R3, 3R2;  (2) 1R2, 1R3, 2R3, 3R2;  (3) 2R1, 2R3, 1R3, 3R1;  (4) 3R1, 3R2, 1R2, 2R1;  (5) 2R1, 3R1, 2R3, 3R2; (6) 1R2, 3R2, 1R3, 3R1;  (7) 1R3, 2R3, 1R2, 2R1;  (8) 1R2, 2R3, 1R3;  (9) 1R3, 3R2, 1R2;  (10) 2R1, 1R3, 2R3;  (11) 2R3, 3R1, 2R1;  (12) 3R1, 1R2, 3R2;  (13) 3R2, 2R1, 3R1; (the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity).
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  1,   12,    13;
  0,  1,   61,   106,    75;
  0,  1,  310,  1105,  1035,   541;
  0,  1, 1821, 12075, 16025, 11301, 4683;
  ...

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A057427, A218092, A218093, A218094, A218095, A218096, A218097, A218098, A218099, A218091.
Main diagonal and lower diagonals give: A000670, A218111, A218112, A218103, A218104, A218105, A218106, A218107, A218108, A218109, A218110.
Row sums are in A052880.
T(2n,n) gives A261238.
Cf. A135302.

t:= proc(k) option remember; `if`(k<0, 0,
      unapply(exp(add(x^m/m!*t(k-m)(x), m=1..k)), x))
    end:
tt:= proc(k) option remember;
       unapply((t(k)-t(k-1))(x), x)
     end:
T:= proc(n, k) option remember;
      coeff(series(tt(k)(x), x, n+1), x, n)*n!
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k - m, x], {m, 1, k}]]; (* a = A135302 *) a[0, 0] = 1; a[, 0] = 0; a[n, k_] := SeriesCoefficient[f[k, x], {x, 0, n}]*n!; t[n_, 0] := a[n, 0]; t[n_, k_] := a[n, k] - a[n, k-1]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2013, after A135302 *)

T(n,0) = A135302(n,0), T(n,k) = A135302(n,k) - A135302(n,k-1) for k>0.

E.g.f. of column k=0: tt_0(x) = 1, e.g.f. of column k>0: tt_k(x) = t_k(x) -t_{k-1}(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.

A135302 Square array of numbers A(n,k) (n>=0, k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= k for all x, read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 13, 4, 1, 1, 0, 1, 62, 26, 4, 1, 1, 0, 1, 311, 168, 26, 4, 1, 1, 0, 1, 1822, 1416, 243, 26, 4, 1, 1, 0, 1, 11593, 13897, 2451, 243, 26, 4, 1, 1, 0, 1, 80964, 153126, 29922, 2992, 243, 26, 4, 1, 1, 0, 1, 608833, 1893180, 420841, 41223, 2992, 243, 26, 4, 1, 1

Offset: 0

Alois P. Heinz, Dec 04 2007

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

			Table A(n,k) begins:
  1, 1,   1,    1,    1,    1, ...
  0, 1,   1,    1,    1,    1, ...
  0, 1,   4,    4,    4,    4, ...
  0, 1,  13,   26,   26,   26, ...
  0, 1,  62,  168,  243,  243, ...
  0, 1, 311, 1416, 2451, 2992, ...

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000012, A135312, A210911, A210912, A210913, A210914, A210915, A210916, A210917, A210918.
Main diagonal gives A052880.
A(n,n)-A(n,n-1) gives A000670.
Cf. A135313.

t:= proc(k) option remember; `if`(k<0, 0,
       unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
A:= proc(n, k) option remember;
      coeff(series(t(k)(x), x, n+1), x, n) *n!
    end:
seq(seq(A(d-i, i), i=0..d), d=0..15);

t[0, ] = 1; t[k, x_] := t[k, x] = Exp[Sum[x^m/m!*t[k-m, x], {m, 1, k}]]; a[0, 0] = 1; a[, 0] = 0; a[n, k_] := SeriesCoefficient[t[k, x], {x, 0, n}]*n!; Table[a[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2013, after Maple *)

E.g.f. of column k=0: t_0(x) = 1; e.g.f. of column k>0: t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)).

A(n,k) = Sum_{i=0..k} A135313(n,i).

A349557 E.g.f. satisfies: log(A(x)) = (exp(xA(x)) - 1) A(x).

Original entry on oeis.org

1, 1, 6, 68, 1163, 26787, 778128, 27325321, 1126308870, 53323302708, 2851990661789, 170088808988705, 11192134680722586, 805521092432042573, 62950026461699015998, 5308512876799649771192, 480492707646769163920059, 46464318322169305448661915

Offset: 0

Seiichi Manyama, Nov 21 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..345

Cf. A052880, A349558, A349560.

a[n_] := Sum[(n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 22 2021 *)

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

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

a(n) ~ sqrt(s^3 * (1+s) / (1 + r^2*s^2*(1+s) + r*s*(3 + 2*s))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.1609673785833512641321517974482987852086944930869... and s = 1.597727491873940099115048788232158935283220293884... are real roots of the system of equations exp(r*s)*s = s + log(s), exp(r*s)*(1 + r*s) = 1 + 1/s. - Vaclav Kotesovec, Nov 22 2021

A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

Original entry on oeis.org

1, 1, 5, 40, 447, 6421, 112726, 2338799, 55990213, 1519122598, 46066158817, 1543974969769, 56677405835276, 2261488166321697, 97455090037460785, 4510770674565054000, 223183550978156866507, 11755122645815049275521, 656670295411196201190366, 38779502115371642484125915, 2413908564514961126280655257

Offset: 0

Ilya Gutkovskiy, Feb 08 2017

Stirling transform of A000312.

			E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 40*x^3/3! + 447*x^4/4! + 6421*x^5/5! + 112726*x^6/6! + ...

G. C. Greubel, Table of n, a(n) for n = 0..375
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's MathWorld, Stirling Transform

Cf. A000312, A000670, A038052, A048802, A052880, A308490, A308491.

b:= proc(n, m) option remember;
     `if`(n=0, m^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2021

Range[0, 20]! CoefficientList[Series[1/(1 + ProductLog[1 - Exp[x]]), {x, 0, 20}], x]
Join[{1}, Table[Sum[StirlingS2[n, k] k^k, {k, 1, n}], {n, 1, 20}]]

x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(1-exp(x))))) \\ G. C. Greubel, Nov 12 2017

a(0) = 1, a(n) = Sum_{k=1..n} Stirling2(n,k)*k^k.

a(n) ~ n^n / (sqrt(1+exp(1)) * (log(1+exp(-1)))^(n+1/2) * exp(n)). - Vaclav Kotesovec, Feb 17 2017

A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

Original entry on oeis.org

1, 1, 0, 2, -9, 72, -710, 8563, -121814, 1997502, -37097739, 769687954, -17644355410, 442894514285, -12081668234012, 355889274553166, -11258683640579857, 380701046875217492, -13702507978018209458, 523049811008797507683, -21105565578064063658754

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..399
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A120980, A349561, A349585.

Cf. A008277, A052880, A349524, A349525.

b:= proc(n, m) option remember; `if`(n=0,
     (1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

a(n) = Sum_{k=0..n} (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(exp(x) - 1) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) ~ -(-1)^n * sqrt(exp(1) - 1) * n^(n-1) / (exp(n+1) * (1 - log(exp(1) - 1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 2, 3, 40, 185, 2556, 22057, 349616, 4519377, 83642860, 1439639201, 31015493928, 663158322697, 16468280168900, 418772642545545, 11847925722273376, 348085509493265825, 11091199095506163420, 368912674236287743633, 13099432280183074041560

Offset: 0

Seiichi Manyama, Jul 18 2022

nonn

Cf. A052880, A355781, A356785.

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

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

a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 28 2022

E.g.f.: exp( -LambertW(x * (1 - exp(x))) ).
E.g.f.: LambertW(x * (1 - exp(x))) / (x * (1 - exp(x))).
a(n) ~ sqrt(1 + exp(1+r)*r^2) * n^(n-1) / (exp(n-1) * r^n), where r = 0.528399250336668412340528181936966763473482889289226687323... is the root of the equation exp(1+r) - exp(1) = 1/r. - Vaclav Kotesovec, Jul 21 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - Seiichi Manyama, Aug 28 2022

A349525 a(n) = Sum_{k=0..n} (3k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 8, 122, 2847, 90112, 3611162, 175352515, 10009442658, 656934750150, 48744407335597, 4035143806865514, 368706775967717518, 36861117438297883213, 4002400525694764367212, 469049713401827161071110, 59010099414303871987517111, 7932542361585921797125908876

Offset: 0

Vaclav Kotesovec, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..331

Cf. A000110, A008277, A052880, A349505, A349524.

b:= proc(n, m) option remember; `if`(n=0,
     (3*m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

Table[Sum[(3*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[(-LambertW[3*(-E^x + 1)]/(3*(E^x - 1)))^(1/3), {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = sum(k=0, n, (3*k+1)^(k-1)*stirling(n, k, 2)); \\ Seiichi Manyama, Nov 20 2021

N=20; x='x+O('x^N); Vec(sum(k=0, N, (3*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Nov 20 2021

E.g.f.: (-LambertW(3*(-exp(x) + 1)) / (3*(exp(x) - 1)))^(1/3).
E.g.f.: exp(-LambertW(3 - 3*exp(x))/3).
a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(3*exp(1))) and c = exp(1/3) * sqrt((1 + 3*exp(1)) * log(1 + 1/(3*exp(1))) / (2*Pi))/3 = 0.190981550465823640438134269765128596177617920807463710992027181154754728...
a(n) ~ sqrt(1 + 3*exp(1)) * n^(n-1) / (3*exp(n - 1/3) * log(1 + 1/(3*exp(1)))^(n - 1/2)).
E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^3.
G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x). - Seiichi Manyama, Nov 20 2021

A349524 a(n) = Sum_{k=0..n} (2k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 6, 65, 1059, 23232, 642859, 21507733, 844701160, 38108248719, 1942394699283, 110401966739110, 6923805346540685, 474957822716470901, 35377953843680999326, 2843665890900123673997, 245340865605247369255751, 22614510471168438300336440, 2217985444621941684970200607

Offset: 0

Vaclav Kotesovec, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..348

Cf. A000110, A008277, A052880, A349504, A349525.

b:= proc(n, m) option remember; `if`(n=0,
     (2*m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

Table[Sum[(2*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Sqrt[-LambertW[2*(-E^x + 1)]/(2*(E^x - 1))], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = sum(k=0, n, (2*k+1)^(k-1)*stirling(n, k, 2)); \\ Seiichi Manyama, Nov 20 2021

N=20; x='x+O('x^N); Vec(sum(k=0, N, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Nov 20 2021

E.g.f.: sqrt(-LambertW(2*(-exp(x) + 1)) / (2*(exp(x) - 1))).
E.g.f.: exp(-LambertW(2 - 2*exp(x))/2).
a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(2*exp(1))) and c = sqrt(exp(1) * (1 + 2*exp(1)) * log(1 + 1/(2*exp(1))) / (2*Pi))/2 = 0.3428481589262346912499652905097648170872882109000404115070292580887155335...
a(n) ~ sqrt(1 + 2*exp(1)) * n^(n-1) / (2*exp(n - 1/2) * log(1 + 1/(2*exp(1)))^(n - 1/2)).
E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^2.
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x). - Seiichi Manyama, Nov 20 2021

A356973 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^3) - 1) * A(x).

Original entry on oeis.org

1, 1, 10, 206, 6555, 283777, 15577332, 1036984027, 81191314678, 7311591070938, 744577308572189, 84608911909469235, 10613728203840498210, 1456899252646375490851, 217215453964895439271178, 34956361099228031471844962, 6039398076840098381458042875

Offset: 0

Seiichi Manyama, Sep 07 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..319

Cf. A052880, A349557, A356972.

Cf. A216136, A356960.

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

a(n) = Sum_{k=0..n} (3*n+k+1)^(k-1) * Stirling2(n,k).

cp's OEIS Frontend

A033917 Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1.

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A135313 Triangle of numbers T(n,k) (n>=0, n>=k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where k=max_{x}(|{y : xRy}|), read by rows.

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A135302 Square array of numbers A(n,k) (n>=0, k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= k for all x, read by antidiagonals.

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A349557 E.g.f. satisfies: log(A(x)) = (exp(x*A(x)) - 1) * A(x).

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A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

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A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

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A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).

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A349557 E.g.f. satisfies: log(A(x)) = (exp(xA(x)) - 1) A(x).

A349525 a(n) = Sum_{k=0..n} (3k+1)^(k-1) Stirling2(n,k).

A349524 a(n) = Sum_{k=0..n} (2k+1)^(k-1) Stirling2(n,k).