A004212 -id:A004212 - 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

A004213 Shifts one place left under 4th-order binomial transform.

Original entry on oeis.org

1, 1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, 1625661357673, 29905322979421, 580513190237573, 11850869542405409, 253669139947767777, 5678266212792053029, 132607996474971041789, 3224106929536557918697

Offset: 0

N. J. A. Sloane

Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+4 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=4, otherwise F(k+1)=F(k); see example and Fxtbook link. - Joerg Arndt, Apr 30 2011

			Restricted growth strings: a(0)=1 corresponds to the empty string, a(1)=1 to [0],
a(2)=3 to [00], [01], [02], [03], and [04], a(3) = 29 to
       RGS          F
.1:  [ 0 0 0 ]    [ 0 0 0 ]
.2:  [ 0 0 1 ]    [ 0 0 0 ]
.3:  [ 0 0 2 ]    [ 0 0 0 ]
.4:  [ 0 0 3 ]    [ 0 0 0 ]
.5:  [ 0 0 4 ]    [ 0 0 4 ]
.6:  [ 0 1 0 ]    [ 0 0 0 ]
.7:  [ 0 1 1 ]    [ 0 0 0 ]
.8:  [ 0 1 2 ]    [ 0 0 0 ]
.9:  [ 0 1 3 ]    [ 0 0 0 ]
10:  [ 0 1 4 ]    [ 0 0 4 ]
11:  [ 0 2 0 ]    [ 0 0 0 ]
12:  [ 0 2 1 ]    [ 0 0 0 ]
13:  [ 0 2 2 ]    [ 0 0 0 ]
14:  [ 0 2 3 ]    [ 0 0 0 ]
15:  [ 0 2 4 ]    [ 0 0 4 ]
16:  [ 0 3 0 ]    [ 0 0 0 ]
17:  [ 0 3 1 ]    [ 0 0 0 ]
18:  [ 0 3 2 ]    [ 0 0 0 ]
19:  [ 0 3 3 ]    [ 0 0 0 ]
20:  [ 0 3 4 ]    [ 0 0 4 ]
21:  [ 0 4 0 ]    [ 0 4 4 ]
22:  [ 0 4 1 ]    [ 0 4 4 ]
23:  [ 0 4 2 ]    [ 0 4 4 ]
24:  [ 0 4 3 ]    [ 0 4 4 ]
25:  [ 0 4 4 ]    [ 0 4 4 ]
26:  [ 0 4 5 ]    [ 0 4 4 ]
27:  [ 0 4 6 ]    [ 0 4 4 ]
28:  [ 0 4 7 ]    [ 0 4 4 ]
29:  [ 0 4 8 ]    [ 0 4 8 ]
[_Joerg Arndt_, Apr 30 2011]

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..66
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; 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]
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A075499 (row sums).

A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A005011 (s(k)<=F(k)+5), A000110 (s(k)<=F(k)+1). - Joerg Arndt, Apr 30 2011

A004213 := proc(n)
    add(4^(n-m)*combinat[stirling2](n,m),m=0..n) ;
end proc:
seq(A004213(n),n=0..30) ; # R. J. Mathar, Aug 20 2022

Table[4^n BellB[n, 1/4], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n):=if n=0 then 1 else sum(4^(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(4*x))); /* =  1 + x + 5/2*x^2 + 29/6*x^3 + 67/8*x^4 + ... */
/* egf=exp(1/4*(exp(4*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

a(n) = Sum_{m=0..n} 4^(n-m)*Stirling2(n, m).
E.g.f.: exp((exp(4*x)-1)/4).
O.g.f. A(x) satisfies A'(x)/A(x) = e^(4x).
E.g.f.: exp(Integral_{t = 0..x} exp(4*t)). - Joerg Arndt, Apr 30 2011
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1-4*j*x). - Joerg Arndt, Apr 30 2011
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^(-1/4)*4^{n-1}*f_n(1/4). - Milan Janjic, May 30 2008
a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (4,4,4,...) is appended to the right of Pascal's triangle:
  1, 4, 0, 0, 0, ...
  1, 1, 4, 0, 0, ...
  1, 2, 1, 4, 0, ...
  1, 3, 3, 1, 4, ...
  ... - Gary W. Adamson, Jul 29 2011
G.f. satisfies A(x) = 1 + x/(1 - 4*x)*A(x/(1 - 4*x)). a(n) = Sum_{k = 1..n} 4^(n-k)*binomial(n-1,k-1)*a(k-1), n > 0, a(0) = 1. - Vladimir Kruchinin, Nov 28 2011 [corrected by Ilya Gutkovskiy, May 02 2019]
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-4*k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 24 2013
G.f.: (G(0) - 1)/(1+x) where G(k) =  1 + 1/(1-4*k*x)/(1-x/(x+1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-x-4*x*k)*(1-5*x-4*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
a(n) = exp(-1/4) * Sum_{k>=0} 4^(n-k) * k^n / k!. - Vaclav Kotesovec, Jul 15 2021
a(n) ~ 4^n * n^n * exp(n/LambertW(4*n) - 1/4 - n) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^n). - Vaclav Kotesovec, Jul 15 2021
From Peter Bala, Jun 29 2024: (Start)
a(n) = exp(-1/4)*Sum_{n >= 0} (4*n)^k/(n!*4^n).
Touchard's congruence holds: for odd prime p, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,.... In particular, a(p) == 2 (mod p) for odd prime p. See A004211. (End)

A005011 Shifts one place left under 5th-order binomial transform.

Original entry on oeis.org

1, 1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, 11458179765541, 249255304141006, 5725640423174901, 138407987170952351, 3510263847256823056

Offset: 0

N. J. A. Sloane, Simon Plouffe

Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+5 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=5, otherwise F(k+1)=F(k); see examples in A004211, A004212, and A004213, and Fxtbook link. [Joerg Arndt, Apr 30 2011]

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..66
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
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]
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A075500 (row sums).

A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A004213 (s(k)<=F(k)+4), A000110 (s(k)<=F(k)+1). - Joerg Arndt, Apr 30 2011

Table[5^n BellB[n, 1/5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

x='x+O('x^66);
egf=exp(intformal(exp(5*x))); /* =  1 + x + 3*x^2 + 41/6*x^3 + 331/24*x^4 + ... */
/* egf=exp(1/5*(exp(5*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

a(n) = Sum_{m=0..n} 5^(n-m)*Stirling2(n, m), n >= 0.
E.g.f.: exp((exp(5*x)-1)/5).
O.g.f. A(x) satisfies A'(x)/A(x) = exp(5*x).
E.g.f.: exp(Integral_{t = 0..x} exp(5*t)). - Joerg Arndt, Apr 30 2011
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1-5*j*x). - Joerg Arndt, Apr 30 2011
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^(-1/5)*5^{n-1}*f_n(1/5). - Milan Janjic, May 30 2008
a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (5,5,5,...) is appended to the right of Pascal's triangle:
  1, 5, 0, 0, 0, ...
  1, 1, 5, 0, 0, ...
  1, 2, 1, 5, 0, ...
  1, 3, 3, 1, 5, ...
  ... - Gary W. Adamson, Jul 29 2011
G.f.: T(0)/(1-x), where T(k) = 1 - 5*x^2*(k+1)/( 5*x^2*(k+1) - (1-x-5*x*k)*(1-6*x-5*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 - 5*x))/(1 - 5*x). - Ilya Gutkovskiy, May 03 2019
a(n) ~ 5^n * n^n * exp(n/LambertW(5*n) - 1/5 - n) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^n). - Vaclav Kotesovec, Jul 15 2021
From Peter Bala, Jun 29 2024: (Start)
a(n) = exp(-1/5)*Sum_{n >= 0} (5*n)^k/(n!*5^n).
Touchard's congruence holds: for all primes p != 5, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,.... In particular, a(p) == 2 (mod p) for all primes p != 5. See A004211. (End)

A075498 Stirling2 triangle with scaled diagonals (powers of 3).

Original entry on oeis.org

1, 3, 1, 9, 9, 1, 27, 63, 18, 1, 81, 405, 225, 30, 1, 243, 2511, 2430, 585, 45, 1, 729, 15309, 24381, 9450, 1260, 63, 1, 2187, 92583, 234738, 137781, 28350, 2394, 84, 1, 6561, 557685, 2205225, 1888110, 563031, 71442, 4158, 108, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(3*z) - 1)*x/3) - 1.
Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013
Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			[1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
From _Philippe Deléham_, Feb 13 2013: (Start)
Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1;
  0,   1;
  0,   3,   1;
  0,   9,   9,   1;
  0,  27,  63,  18,   1;
  0,  81, 405, 225,  30,   1;
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.

Cf. A075497, A075499.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
rows = 9;
t = Table[3^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (3^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 3*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
From Peter Bala, Jan 13 2018: (Start)
Dobinski-type formulas for row polynomials R(n,x):
R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)

A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

Original entry on oeis.org

1, 1, -2, 1, 19, -128, 379, 1549, -32600, 261631, -845909, -10713602, 237695149, -2513395259, 11792378662, 151915180429, -4826456213273, 70741388773960, -558513179369297, -2833805536521839, 200720356696607416, -4256279445015662093, 54120395442382043743, -173423789950999240226

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..495
Eric Weisstein's World of Mathematics, Bell Polynomial

Column k=3 of A309386.

Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.

a:=series(exp((1 - exp(-3*x))/3), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = Sum[(-3)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* Peter Luschny, Aug 20 2018 *)

a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-3)^n BellPolynomial_n(-1/3). - Peter Luschny, Aug 20 2018

A049425 Row sums of triangle A049404.

Original entry on oeis.org

1, 1, 3, 9, 33, 141, 651, 3333, 18369, 108153, 678771, 4495041, 31324833, 228803589, 1744475643, 13852095741, 114235118721, 976176336753, 8627940414819, 78726234866553, 740440277799201, 7168107030092541, 71331617341611243, 728811735008913909, 7637128289949856833, 81995144342947130601

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..616 (terms 0..200 from Vincenzo Librandi)
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, J. Integer Seqs., Vol. 20 (2017), #17.8.2.

Column k=2 of A293991.

Cf. A004212.

Table[n!*SeriesCoefficient[E^(x+x^2+(x^3)/3),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)

/* for b(n) = a(n+1) */
b(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+2,n-2*i-k)/3^i,i,0,k),k,0,n);
makelist(b(n),n,0,24);  /* Emanuele Munarini, Oct 20 2014 */

x='x+O('x^66); Vec(serlaplace(exp(x+x^2+(x^3)/3))) \\ Joerg Arndt, May 04 2013

E.g.f.: exp(x+x^2+(x^3)/3).
a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(3*j,n) * (-1)^(k-j)/(3^k * (k-j)!*j!))). [Vladimir Kruchinin, Feb 07 2011]
Conjecture: -a(n) +a(n-1) +(2*n-2)*a(n-2) + (2-3*n+n^2)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ exp(n^(2/3)+n^(1/3)/3-2*n/3-2/9)*n^(2*n/3)/sqrt(3)*(1+59/(162*n^(1/3))). - Vaclav Kotesovec, Oct 08 2012
From Emanuele Munarini, Oct 20 2014: (Start)
Recurrence: a(n+3) = a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n).
It derives from the differential equation for the e.g.f.: A'(x) = (1+2*x+x^2)*A(x).
So, the above conjecture is true.
b(n) = a(n+1) = sum((n!/k!)*sum(bin(k,i)*bin(k-i+2,n-2*i-k)/3^i,i=0..k),k=0..n).
E.g.f. for b(n) = a(n+1): (1+t)^2*exp(t+t^2+t^3/3).
(End)
a(n) = Sum_{k=0..n} Stirling1(n,k) * A004212(k). - Seiichi Manyama, Jan 31 2024
a(n) = (1/exp(1/3)) * n! * Sum_{k>=0} binomial(3*k,n)/(3^k * k!). - Seiichi Manyama, Jan 18 2025

cp's OEIS Frontend

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

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A004213 Shifts one place left under 4th-order binomial transform.

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A005011 Shifts one place left under 5th-order binomial transform.

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A075498 Stirling2 triangle with scaled diagonals (powers of 3).

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A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

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A049425 Row sums of triangle A049404.

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A075509 Shifts one place left under 10th-order binomial transform.

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