A055882 -id:A055882 - cp's OEIS frontend

A004211 Shifts one place left under 2nd-order binomial transform.

1, 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, 5120905441, 56878092067, 664920021819, 8155340557697, 104652541401025, 1401572711758403, 19546873773314571, 283314887789276721, 4259997696504874817, 66341623494636864963

Offset: 0

N. J. A. Sloane

Equals the eigensequence of A038207, the square of Pascal's triangle. - Gary W. Adamson, Apr 10 2009
The g.f. of the second binomial transform is 1/(1-2x-x/(1-2x/(1-2x-x/(1-4x/(1-2x-x/(1-6x/(1-2x-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009
Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+2 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=2, otherwise F(k+1)=F(k); see example and Fxtbook link. - Joerg Arndt, Apr 30 2011
It appears that the infinite set of "Shifts 1 place left under N-th order binomial transform" sequences has a production matrix of:
  1, N, 0, 0, 0, ...
  1, 1, N, 0, 0, ...
  1, 2, 1, N, 0, ...
  1, 3, 3, 1, N, ...
  ... (where a diagonal of (N,N,N,...) is appended to the right of Pascal's triangle). a(n) in each sequence is the upper left term of M^n such that N=1 generates A000110, then (N=2 - A004211), (N=3 - A004212), (N=4 - A004213), (N=5 - A005011), ... - Gary W. Adamson, Jul 29 2011
Number of "unlabeled" hierarchical orderings on set partitions of {1..n}, see comments on A034691. - Gus Wiseman, Mar 03 2016
From Lorenzo Sauras Altuzarra, Jun 17 2022: (Start)
Number of n-variate noncontradictory conjunctions of logical equalities of literals (modulo logical equivalence).
Equivalently, number of n-variate noncontradictory Krom formulas with palindromic truth-vector (modulo logical equivalence).
a(n) <= A109457(n). (End)

			From _Joerg Arndt_, Apr 30 2011: (Start)
Restricted growth strings: a(0)=1 corresponds to the empty string;
a(1)=1 to [0]; a(2)=3 to [00], [01], and [02]; a(3) = 11 to
        RGS           F
[ 1]  [ 0 0 0 ]    [ 0 0 0 ]
[ 2]  [ 0 0 1 ]    [ 0 0 0 ]
[ 3]  [ 0 0 2 ]    [ 0 0 2 ]
[ 4]  [ 0 1 0 ]    [ 0 0 0 ]
[ 5]  [ 0 1 1 ]    [ 0 0 0 ]
[ 6]  [ 0 1 2 ]    [ 0 0 2 ]
[ 7]  [ 0 2 0 ]    [ 0 2 2 ]
[ 8]  [ 0 2 1 ]    [ 0 2 2 ]
[ 9]  [ 0 2 2 ]    [ 0 2 2 ]
[10]  [ 0 2 3 ]    [ 0 2 2 ]
[11]  [ 0 2 4 ]    [ 0 2 4 ]. (End)
From _Lorenzo Sauras Altuzarra_, Jun 17 2022: (Start)
The 11 trivariate noncontradictory conjunctions of logical equalities of literals are (x <-> y) /\ (y <-> z), (~ x <-> y) /\ (y <-> z), (x <-> ~ y) /\ (~ y <-> z), (x <-> y) /\ (y <-> ~ z), (x <-> y) /\ (z <-> z), (~ x <-> y) /\ (z <-> z), (x <-> z) /\ (y <-> y), (~ x <-> z) /\ (y <-> y), (y <-> z) /\ (x <-> x), (~ y <-> z) /\ (x <-> x), and (x <-> x) /\ (y <-> y) /\ (z <-> z) (modulo logical equivalence).
The third complete Bell polynomial is x^3 + 3 x y + z; and note that (2^0)^3 + 3*2^0*2^1 + 2^2 = 11.
The truth-vector of (x <-> y) /\ (y <-> z), 10000001, is palindromic. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, On the Limiting Vacillating Tableaux for Integer Sequences, arXiv:2208.13091 [math.CO], 2022.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 22, 29.
David Callan (Proposer), Problem 11567, Amer. Math. Monthly, 118 (2011), 371-378.
Agustina Czenky, Unoriented 2-dimensional TQFTs and the category Rep(S_t wr Z_2), arXiv:2306.08826 [math.QA], 2023. See p. 40.
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121v3 [math.CO], 2001.
Adam M. Goyt and Lara K. Pudwell, Solution to Monthly Problem 11567, 14 May 2011.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 20.
Janusz Milek, Quantum Implementation of Risk Analysis-relevant Copulas, arXiv:2002.07389 [stat.ME], 2020.
N. J. A. Sloane, Transforms

Cf. A075497 (row sums).
Cf. A038207.
Cf. A000110 (RGS where s(k) <= F(k) + 1), A004212 (RGS where s(k) <= F(k) + 3), A004213 (s(k) <= F(k) + 4), A005011 (s(k) <= F(k) + 5), A005012 (s(k) <= F(k) + 6), A075506 (s(k) <= F(k) + 7), A075507 (s(k) <= F(k) + 8), A075508 (s(k) <= F(k) + 9), A075509 (s(k) <= F(k) + 10).
Cf. A000587, A009235, A317996, A318179 - A318181.
Main diagonal of A261275.
Cf. A034691, A075729, A109457.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*2^(j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021
# second Maple program:
a:= n -> CompleteBellB(n, seq(2^k, k=0..n)):
seq(a(n), n=0..23);  # Lorenzo Sauras Altuzarra, Jun 17 2022

Table[ Sum[ StirlingS2[ n, k ] 2^(-k+n), {k, n} ], {n, 16} ] (* Wouter Meeussen *)
Table[2^n BellB[n, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n):=if n=0 then 1 else sum(2^(n-k)*binomial(n-1,k-1)*a(k-1),k,1,n); /* Vladimir Kruchinin, Nov 28 2011 */

x='x+O('x^66);
egf=exp(intformal(exp(2*x))); /* = 1 + x + 3/2*x^2 + 11/6*x^3 + ... */
/* egf=exp(1/2*(exp(2*x)-1)) */ /* alternative e.g.f. */
Vec(serlaplace(egf))  /* Joerg Arndt, Apr 30 2011 */

def A004211(n): return sum(2^(n-k)*stirling_number2(n, k) for k in (0..n))
print([A004211(n) for n in range(21)]) # Peter Luschny, Apr 15 2020

E.g.f. A(x) satisfies A'(x)/A(x) = e^(2x).
E.g.f.: exp(sinh(x)*exp(x)) = exp(Integral_{t = 0..x} exp(2*t)) = exp((exp(2*x)-1)/2). - Joerg Arndt, Apr 30 2011 and May 13 2011
a(n) = Sum_{k=0..n} 2^(n-k)*Stirling2(n, k). - Emeric Deutsch, Feb 11 2002
G.f.: Sum_{k >= 0} x^k/Product_{j = 1..k} (1-2*j*x). - Ralf Stephan, Apr 18 2004
Stirling transform of A000085. - Vladeta Jovovic May 14 2004
O.g.f.: A(x) = 1/(1-x-2*x^2/(1-3*x-4*x^2/(1-5*x-6*x^2/(1-... -(2*n-1)*x-2*n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/2)*2^{n-1}*f_n(1/2). - Milan Janjic, May 30 2008
G.f.: 1/(1-x/(1-2x/(1-x/(1-4x/(1-x/(1-6x/(1-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009
a(n) = upper left term in M^n, M = an infinite square production matrix with an appended diagonal of (2,2,2,...) to the right of Pascal's triangle:
  1, 2, 0, 0, 0, ...
  1, 1, 2, 0, 0, ...
  1, 2, 1, 2, 0, ...
  1, 3, 3, 1, 2, ...
  ... - Gary W. Adamson, Jul 29 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A000110. - Peter Bala, Nov 25 2011
G.f. A(x) satisfies A(x)=1+x/(1-2*x)*A(x/(1-2*x)), a(n) = Sum_{k=1..n} binomial(n-1,k-1)*2^(n-k)*a(k-1), a(0)=1. - Vladimir Kruchinin, Nov 28 2011 [corrected by Ilya Gutkovskiy, May 02 2019]
From Peter Bala, May 16 2012: (Start)
Recurrence equation: a(n+1) = Sum_{k = 0..n} 2^(n-k)*C(n,k)*a(k).
Written umbrally this is a(n+1) = (a + 2)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a+2) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a-2)*(a-4)*...*(a-2*(n-1)) = 1 with a(0) = 1. Compare with the Bell numbers B(n) = A000110(n), which satisfy the umbral recurrence B*(B-1)*...*(B-(n-1)) = 1 with B(0) = 1. Cf. A009235.
Touchard's congruence holds: for odd prime p, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) == 2 (mod p) for odd prime p. (End)
G.f.: (2/E(0) - 1)/x  where E(k) = 1 + 1/(1 + 2*x/(1 - 4*(k+1)*x/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: (1/E(0)-1)/x  where E(k) = 1 - x/(1 + 2*x - 2*x*(k+1)/E(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Sep 21 2012
a(n) = -1 + 2*Sum_{k=0..n} C(n,k)*A166922(k). - Peter Luschny, Nov 01 2012
G.f.: G(0)- 1/x where G(k) = 1 - (4*x*k-1)/(x - x^4/(x^3 - (4*x*k-1)*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 08 2013.
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-2*k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: -G(0) where G(k) = 1 + 2*(1-k*x)/(2*k*x - 1 - x*(2*k*x - 1)/(x - 2*(1-k*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2013
G.f.: 1/Q(0), where Q(k) = 1 - x/(1 - 2*x*(2*k+1)/( 1 - x/(1 - 4*x*(k+1)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013
G.f.: 1 + x/Q(0), where Q(k)= 1 - x*(2*k+3) - x^2*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 05 2013
For n > 0, a(n) = exp(-1/2)*Sum_{k > 0} (2*k)^n/(k!*2^k). - Vladimir Reshetnikov, May 09 2013
G.f.: -(1+(2*x+1)/G(0))/x, where G(k)= 2*x*k - x - 1 - 2*(k+1)*x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 20 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
Sum_{k=0..n} C(n,k)*a(k)*a(n-k) = 2^n*Bell(n) = A055882(n). - Vaclav Kotesovec, Apr 03 2016
a(n) ~ 2^n * n^n * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^n). - Vaclav Kotesovec, Jan 07 2019, simplified Oct 01 2022
a(n) = B_n(2^0, ..., 2^(n - 1)), where B_n(x_1, ..., x_n) is the n-th complete Bell polynomial. - Lorenzo Sauras Altuzarra, Jun 17 2022

A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block.

Original entry on oeis.org

1, 2, 12, 104, 1200, 17312, 299712, 6053504, 139733760, 3628677632, 104701504512, 3323151509504, 115063060869120, 4316023589937152, 174347763227738112, 7545919601962287104, 348366745238330081280, 17087957176042900815872, 887497598764802460352512

Offset: 0

Geoffrey Critzer, Sep 16 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, Generalized Barred Preferential Arrangements, Transactions on Combinatorics (2022).
Sithembele Nkonkobe, Venkat Murali, and Béata Bényi, Generalised Barred Preferential Arrangements, arXiv:1907.08944 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A000556, A000670, A006153, A055882, A080253.

a := n -> 2^(n-1)*(polylog(-n, 1/2)+`if`(n=0,1,0)):
seq(round(evalf(a(n),32)), n=0..18); # Peter Luschny, Nov 03 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 04 2019

nn=25;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[1/(1-a),{x,0,nn}],x]
Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[2]] + PolyLog[-n, -Sqrt[2]])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

a(n) = 2^(n-1)*(polylog(-n, 1/2) + 0^n); \\ Michel Marcus, May 30 2018

def A216794(n):
    return 2^n*add(add((-1)^(j-i)*binomial(j,i)*i^n for i in range(n+1)) for j in range(n+1))
[A216794(n) for n in range(18)] # Peter Luschny, Jul 22 2014

E.g.f.: 1/(2 - exp(2*x)).
E.g.f.: 1 + 2*x/(G(0) - 2*x) where G(k) = 2*k+1 - x*2*(2*k+1)/(2*x + (2*k+2)/(1 + 2*x/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
E.g.f.: 1 + 2*x/( G(0) - 2*x ) where G(k) = 1 - 2*x/(1 + (1*k+1)/G(k+1)); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+2)/( 1 - 4*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * (2/log(2))^n/log(4). - Vaclav Kotesovec, Sep 24 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 8*x^2*(k+1)^2/( 8*x^2*(k+1)^2 - (1-2*x-6*x*k)*(1-8*x-6*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
From Vladimir Reshetnikov, Oct 31 2015: (Start)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) + Li_{-n}(-sqrt(2)))/4, where Li_n(x) is the polylogarithm.
Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*a(n) + A080253(n)*sqrt(2)).
(End)
a(n) = 2^(n-1)*(Li_{-n}(1/2) + 0^n) with 0^0=1. - Peter Luschny, Nov 03 2015
From Peter Bala, Oct 18 2023: (Start)
a(n) = 2^n * A000670(n)
Inverse binomial transform of A080253.
The sequence is the first column of the array (2*I - P^2)^(-1), where P denotes Pascal's triangle A007318. (End)

A143405 Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.

Original entry on oeis.org

1, 1, 4, 17, 89, 552, 3895, 30641, 265186, 2497551, 25373097, 276105106, 3199697517, 39297401197, 509370849148, 6943232742493, 99217486649933, 1482237515573624, 23093484367004715, 374416757914118941, 6304680593346141746, 110063311977033807187

Offset: 0

Alois P. Heinz, Aug 12 2008

nonn

a(n) is the number of the partitions of an n-set where each block is endowed with a nonempty subset. - Emanuele Munarini, Sep 15 2016

			a(2) = 4, because there are 4 forests for 2 labels: {1,2}, {1}{2}, {1}<-2, {2}<-1.

Alois P. Heinz, Table of n, a(n) for n = 0..504
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vaclav Kotesovec, Asymptotics of OEIS A143405, A355291 and the generalization, Jul 12 2022 (this version also includes several figures).
Index entries for sequences related to rooted trees

Cf. A055882, A143395, A143396, A143397, A048993, A008277, A007318, A000110, A355291.

a:= n-> add(add(binomial(n, t)*Stirling2(t, k)*k^(n-t), t=k..n), k=0..n):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 05 2019

CoefficientList[Series[Exp[Exp[t] (Exp[t] - 1)], {t, 0, 12}], t] Range[0, 12]! (* Emanuele Munarini, Sep 15 2016 *)
Table[Sum[Binomial[n, k] 2^k BellB[k] BellB[n - k, -1], {k, 0, n}], {n, 0, 12}] (* Emanuele Munarini, Sep 15 2016 *)
Table[Sum[BellY[n, k, 2^Range[n] - 1], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*stirling(n, k, 2)); \\ Seiichi Manyama, May 14 2022

a(n) = Sum_{k=0..n} Sum_{t=k..n} C(n,t) * Stirling2(t,k)*k^(n-t).
a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k) * Stirling2(k,t)*t^(n-k).
a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t)*t^(k-t).
E.g.f.: exp(exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
a(n) = sum(binomial(n,k)*2^k*bell(k)*S(n-k,-1),k=0..n), where bell(n) are the Bell numbers (A000110) and S(n,x) = sum(Stirling2(n,k)*x^k,k=0..n) are the Stirling (or exponential) polynomials. - Emanuele Munarini, Sep 15 2016
Identity: sum(binomial(n,k)*a(k)*bell(n-k),k=0..n) = 2^n*bell(n). - Emanuele Munarini, Sep 15 2016
a(n) = Sum_{k=0..n} A047974(k) * Stirling2(n,k). - Seiichi Manyama, May 14 2022
a(n) ~ exp(exp(2*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) - (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) - 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 1/8 - n/LambertW(n) + sqrt(n/LambertW(n)))). - Vaclav Kotesovec, Jul 08 2022

A247452 a(n) = 3^n*Bell(n).

Original entry on oeis.org

1, 3, 18, 135, 1215, 12636, 147987, 1917999, 27162540, 416236401, 6848207775, 120206639790, 2239278203277, 44074161731151, 913065539247018, 19843943547060315, 451135755042249987, 10701182793462338052, 264250529777677991751, 6779171511882363638619, 180350988089950776032172

Offset: 0

Karol A. Penson, Sep 17 2014

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..200

Cf. A000110, A055882.

[3^n*Bell(n): n in [0..20]]; // Vincenzo Librandi, Sep 19 2014

Table[3^n BellB[n], {n, 0, 20}] (* Vincenzo Librandi, Sep 19 2014 *)

# Python 3.2 or above required.
from itertools import accumulate
A247452_list, blist, b, n3 = [1,3], [1], 1, 9
for _ in range(2,201):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A247452_list.append(b*n3)
    n3 *= 3 # Chai Wah Wu, Sep 19 2014

a(n) = 3^n*A000110(n).
a(n) = (1/e)*Sum_{k>=0} (3*k)^n/k!. (This is a Dobinski-type formula.)
O.g.f.: (1/e)*Sum_{k>=0} 1/(k!*(1-3*k*z)).
E.g.f.: exp(exp(3*z)-1).
a(n) is the n-th moment of a discrete, positive weight function w(x) consisting of an infinite comb of Dirac delta functions situated at x=3*k, with k = 0, 1, ..., defined as w(x) = (1/e)*Sum_{k>=0} Dirac(x-3*k)/k!.
G.f.: 1/(1-3*x/(1-3*x/(1-3*x/(1-6*x/(1-3*x/(1-9*x/(1-...))))))) (continued fraction). - Philippe Deléham, Sep 18 2014
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - Ilya Gutkovskiy, Jan 16 2020

A355291 Expansion of e.g.f. exp(exp(x)*(exp(x) + 1) - 2).

Original entry on oeis.org

1, 3, 14, 81, 551, 4266, 36803, 348543, 3583484, 39652659, 468970211, 5894584812, 78366374813, 1097537989671, 16136598952718, 248309032411485, 3988468487017379, 66715970326561170, 1159712730763363991, 20909709414253764819, 390374806223071148084, 7534929383736826736007

Offset: 0

Vaclav Kotesovec, Jun 27 2022

nonn

In general, if m > 0, b > d >= 1 and e.g.f. = exp(m*exp(b*x) + r*exp(d*x) + s) then a(n) ~ exp(m*exp(b*z) + r*exp(d*z) + s - n) * (n/z)^(n + 1/2) / sqrt(m*b*(1 + b*z)*exp(b*z) + r*d*(1 + d*z)*exp(d*z)), where z = LambertW(n/m)/b - 1/(d + b/LambertW(n/m) + b^2 * m^(d/b) * n^(1 - d/b) * (1 + LambertW(n/m)) / (d*r*LambertW(n/m)^(2 - d/b))). - Vaclav Kotesovec, Jul 03 2022

In addition, if b/d >=2 then a(n) ~ c * (b*n/LambertW(n/m))^n * exp(n/LambertW(n/m) + r * (n/(m*LambertW(n/m)))^(d/b) - n + s) / sqrt(1 + LambertW(n/m)), where c = 1 for b/d > 2 and c = exp(-r^2/(8*m)) for b/d = 2. - Vaclav Kotesovec, Jul 10 2022

Seiichi Manyama, Table of n, a(n) for n = 0..500
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.

Cf. A001861, A055882, A126390, A143405, A355379.

nmax = 20; CoefficientList[Series[Exp[Exp[2*x] - 2 + Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x)*(exp(x) + 1) - 2))) \\ Michel Marcus, Jun 27 2022

a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * Bell(k) * Bell(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jul 01 2022
a(n) ~ exp(exp(2*z) + exp(z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) + (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) + 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 17/8 - n/LambertW(n) - sqrt(n/LambertW(n)))). - Vaclav Kotesovec, Jul 08 2022

A081066 Even order Taylor expansion coefficients at x=0 of exp(exp(x^2/2)-1), odd order coefficients being equal to zero.

Original entry on oeis.org

1, 1, 6, 75, 1575, 49140, 2110185, 118513395, 8391883500, 728713460475, 75932204473125, 9329869676877750, 1332483237190430325, 218552871240812233125, 40748996386059790578750

Offset: 0

Karol A. Penson, Mar 04 2003

nonn

From Shai Covo (green355(AT)netvision.net.il), Feb 03 2010: (Start)
Let N be a Poisson random variable with parameter (mean) 1, and Y_1,Y_2,... independent Normal(0,1) (standard normal) random variables, independent of N.
Set S=Sum_{i=1..N} Y_i. Then the moment generating function (MGF) of S is given by exp(exp(x^2/2)-1) (i.e., this is the expectation of exp(xS), x real); hence a(n) is the 2n-th moment of S (the odd moments are equal to zero). More generally, if N above has parameter lambda and Y_i above are Normal(0,sigma^2), then the MGF of S is given by exp(lambda*(exp(sigma^2*x^2/2)-1)) and the 2n-th moment of S is given by (2n-1)!!*sigma^(2n)*Sum_{j=0..n} S2(n,j)*lambda^j, where S2(n,j) are the Stirling numbers of the second kind. (End)

S. Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci. 7 (2009), 91-100. [From Shai Covo (green355(AT)netvision.net.il), Feb 03 2010]

Franklin T. Adams-Watters, Jun 18 2008, Table of n, a(n) for n = 0..18

Cf. A000110, A001147.

A055882 := n-> 2^n*combinat[bell](n);
A081066 := n-> A055882(n)*pochhammer(1/2,n);
seq(A081066(i),i=0..14); # Peter Luschny, Nov 07 2011

Table[(2n-1)!!*BellB[n], {n, 0, 14}] (* Jean-François Alcover, May 23 2016, after Vladeta Jovovic *)

In Maple notation: a(n)=evalf(subs(x=0, diff((exp(exp(x^2/2)-1), x$2*n)))), n=1, 2...
a(n) = (2*n-1)!!*Bell(n). - Vladeta Jovovic, May 19 2007
E.g.f.: A(x) = exp(-1)*Sum_{n>=0} (1-2*n*x)^(-1/2)/n!. - Vladeta Jovovic, Feb 05 2008
a(n) = A055882(n)*Pochhammer(1/2, n). - Peter Luschny, Nov 07 2011

A154602 Exponential Riordan array [exp(sinh(x)exp(x)), sinh(x)exp(x)].

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 49, 104, 70, 16, 1, 257, 641, 550, 190, 25, 1, 1539, 4380, 4531, 2080, 425, 36, 1, 10299, 32803, 39515, 22491, 6265, 833, 49, 1, 75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1, 609441, 2337505, 3575820, 2792476, 1192086, 281190, 36204, 2460, 81, 1

Offset: 0

Paul Barry, Jan 12 2009

Triangle T(n,k), read by rows, given by [1,2,1,4,1,6,1,8,1,10,1,12,1,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2009

			Triangle begins
     1;
     1,    1;
     3,    4,    1;
    11,   19,    9,    1;
    49,  104,   70,   16,   1;
   257,  641,  550,  190,  25,  1;
  1539, 4380, 4531, 2080, 425, 36, 1;
Production matrix of this array is
  1,  1,
  2,  3,  1,
  0,  4,  5,  1,
  0,  0,  6,  7,  1,
  0,  0,  0,  8,  9,  1,
  0,  0,  0,  0, 10, 11,  1
with generating function exp(t*x)*(1+t)*(1+2*x).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Columns k=0..3 give A000007, A383203, A383204, A383205.
Cf. A004211 (first column), A256893.
Sums include: A000007 (alternating sign row), A055882 (row sums).

A154602:= func< n,k | (&+[2^(n-j)*Binomial(j,k)*StirlingSecond(n,j): j in [k..n]]) >;
[A154602(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 19 2024

A154602 := (n, k) -> add(2^(n-j) * binomial(j, k) * Stirling2(n, j), j = k..n): for n from 0 to 6 do seq(A154602(n, k), k = 0..n) od; # Peter Luschny, Dec 13 2022

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[Sinh[#] Exp[#]]&, Sinh[#] Exp[#]&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

def A154602(n,k): return sum(2^(n-j)*binomial(j,k)* stirling_number2(n,j) for j in range(k,n+1))
flatten([[A154602(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 19 2024

T(n, 0) = A004211(n).
Sum_{k=0..n} T(n, k) = A055882(n) (row sums).
From Peter Bala, Jun 15 2009: (Start)
T(n,k) = Sum_{i = k..n} 2^(n-i)*binomial(i,k)*Stirling2(n,i).
E.g.f.: exp((t+1)/2*(exp(2*x)-1)) = 1 + (1+t)*x + (3+4*t+t^2)*x^2/2! + ....
Row generating polynomials R_n(x):
R_n(x) = 2^n*Bell(n,(x+1)/2), where Bell(n,x) = Sum_{k = 0..n} Stirling2(n, k)*x^k denotes the n-th Bell polynomial.
Recursion:
R(n+1,x) = (x+1)*(R_n(x) + 2*d/dx(R_n(x))).
(End)
Recurrence: T(n,k) = 2*(k+1)*T(n-1,k+1) + (2*k+1)*T(n-1,k) + T(n-1,k-1). - Emanuele Munarini, Apr 14 2020
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n). - G. C. Greubel, Sep 19 2024
E.g.f. of column k (with leading zeros): f(x)^k * exp(f(x)) / k! with f(x) = (exp(2*x) - 1)/2. - Seiichi Manyama, Apr 19 2025

cp's OEIS Frontend

A004211 Shifts one place left under 2nd-order binomial transform.

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A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block.

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A143405 Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.

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A247452 a(n) = 3^n*Bell(n).

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A355291 Expansion of e.g.f. exp(exp(x)*(exp(x) + 1) - 2).

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A081066 Even order Taylor expansion coefficients at x=0 of exp(exp(x^2/2)-1), odd order coefficients being equal to zero.

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A154602 Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].

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A154602 Exponential Riordan array [exp(sinh(x)exp(x)), sinh(x)exp(x)].

A308543 Expansion of e.g.f. exp(2(exp(2x) - 1)).