A153278 -id:A153278 - cp's OEIS frontend

A003659 Shifts left under Stirling2 transform.

1, 1, 2, 6, 26, 152, 1144, 10742, 122772, 1673856, 26780972, 496090330, 10519217930, 252851833482, 6832018188414, 205985750827854, 6885220780488694, 253685194149119818, 10250343686634687424, 452108221967363310278, 21676762640915055856716

Offset: 1

N. J. A. Sloane, Mira Bernstein

Apart from leading term, number of M-sequences from multicomplexes on at most 4 variables with no monomial of degree more than n+1.
Stirling2 transform of a(n) = [1, 1, 2, 6, 26, ...] is a(n+1) = [1, 2, 6, 26, ...].
Eigensequence of Stirling2 triangle A008277. - Philippe Deléham, Mar 23 2007

S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..330
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]
M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. [_N. J. A. Sloane_, Sep 16 2012]
Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047 [math.CO], 2008. [_Jonathan Vos Post_, Dec 22 2008]
N. J. A. Sloane, Transforms

Cf. A048801.

Cf. A153277, A153278. - Jonathan Vos Post, Dec 22 2008

stirtr:= proc(p)
           proc(n) add(p(k)*Stirling2(n,k), k=0..n) end
         end:
a:= proc(n) option remember; `if`(n<3, 1, aa(n-1)) end:
aa:= stirtr(a):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 22 2012

terms = 21; A[] = 0; Do[A[x] = Normal[Integrate[1 + A[Exp[x] - 1 + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms];
CoefficientList[A[x], x]*Range[0, terms]! // Rest (* Jean-François Alcover, May 23 2012, updated Jan 12 2018 *)

{a(n)=local(A, E); if(n<0, 0, A=O(x); E=exp(x+x*O(x^n))-1; for(m=1, n, A=intformal( subst( 1+A, x, E+x*O(x^m)))); n!*polcoeff(A, n))} /* Michael Somos, Mar 08 2004 */

a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, stirling(i, j, 2)*v[j])); v; \\ Seiichi Manyama, Jun 24 2022

E.g.f. A(x) satisfies A(x)' = 1+A(exp(x)-1).
G.f. satisfies: Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - j*x)). - Ilya Gutkovskiy, May 09 2019
a(1) = 1; a(n+1) = Sum_{k=1..n} Stirling2(n,k) * a(k). - Seiichi Manyama, Jun 24 2022

A363007 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - f^k(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 24, 1, 1, 5, 23, 75, 120, 1, 1, 6, 36, 175, 541, 720, 1, 1, 7, 52, 342, 1662, 4683, 5040, 1, 1, 8, 71, 594, 4048, 18937, 47293, 40320, 1, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 362880, 1, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 3628800

Offset: 0

Seiichi Manyama, May 12 2023

			Square array begins:
    1,   1,    1,    1,    1,     1, ...
    1,   1,    1,    1,    1,     1, ...
    2,   3,    4,    5,    6,     7, ...
    6,  13,   23,   36,   52,    71, ...
   24,  75,  175,  342,  594,   949, ...
  120, 541, 1662, 4048, 8444, 15775, ...

Columns k=0..5 give A000142, A000670, A083355, A099391, A363008, A363009.
Main diagonal gives A363010.
Cf. A153278, A351420, A351429.

T(n, k) = if(k==0, n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));

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

T(n,k) = A153278(k,n) for n >= 1 and k >= 1.

A363008 Expansion of e.g.f. 1/(2 - exp(exp(exp(exp(x) - 1) - 1) - 1)).

Original entry on oeis.org

1, 1, 6, 52, 594, 8444, 143783, 2854261, 64735570, 1651560175, 46814933977, 1459689346911, 49650414218071, 1829560770160335, 72603137881845927, 3086932915850946633, 139999909097319319787, 6746170002325663539844, 344199636595620793896784

Offset: 0

Seiichi Manyama, May 12 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..383

Row p=4 of A153278 (for n>=1).
Column k=4 of A363007.
Cf. A351427.

b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, m!,
      b(m, 0, t-1)), m*b(n-1, m, t)+b(n-1, m+1, t))
    end:
a:= n-> b(n, 0, 4):
seq(a(n), n=0..20);  # Alois P. Heinz, May 12 2023

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

a(n) = T(n,4), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.

A363009 Expansion of e.g.f. 1/(2 - exp(exp(exp(exp(exp(x) - 1) - 1) - 1) - 1)).

Original entry on oeis.org

1, 1, 7, 71, 949, 15775, 313920, 7279795, 192828745, 5744627550, 190131836270, 6921735519110, 274885665920198, 11826225289547024, 547926995688877245, 27199542114163170649, 1440220170795372833970, 81026116511855753816058

Offset: 0

Seiichi Manyama, May 12 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..375

Row p=5 of A153278 (for n>=1).
Column k=5 of A363007.
Cf. A351428.

b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, m!,
      b(m, 0, t-1)), m*b(n-1, m, t)+b(n-1, m+1, t))
    end:
a:= n-> b(n, 0, 5):
seq(a(n), n=0..20);  # Alois P. Heinz, May 12 2023

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

a(n) = T(n,5), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.

A099391 Expansion of e.g.f. 1/(2 - exp(exp(exp(x) - 1) - 1)).

Original entry on oeis.org

1, 1, 5, 36, 342, 4048, 57437, 950512, 17975438, 382424397, 9039989107, 235062317196, 6667866337309, 204905200542916, 6781157167505291, 240446179599065951, 9094120016963808935, 365453749501228063845

Offset: 0

Ralf Stephan, Oct 18 2004

nonn

K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem [J. Phys. A 37 (2004), 3475-3487]

Column k=3 of A363007.
Row p=3 of A153278 (for n>=1).
Cf. A000110, A083355, A130410.

With[{nn=20},CoefficientList[Series[1/(2-Exp[Exp[Exp[x]-1]-1]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 10 2014 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(exp(exp(x)-1)-1)))) \\ Seiichi Manyama, May 12 2023

(1/2) Sum[k=0..inf, k^n/k! * Sum[r=1..inf, e^(-r)r^k/r!*Li(-r, 1/2e) ]], with Li the polylogarithm.
a(n) ~ n! / (2 * (1 + log(2)) * (1 + log(1 + log(2))) * log(1 + log(1 + log(2)))^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(n) = Sum_{k=0..n} Stirling2(n,k) * A083355(k). - Seiichi Manyama, May 12 2023

Definition clarified by Harvey P. Dale, Apr 10 2014

A363010 a(n) = n! * [x^n] 1/(1 - f^n(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, 1, 4, 36, 594, 15775, 618838, 33757864, 2448904188, 228290728635, 26617527649365, 3797508644987398, 651082351708066303, 132130157056046918808, 31333332827346731906130, 8587011712002719806274022, 2693586800519167315881703732, 958983405298849163873718493941

Offset: 0

Seiichi Manyama, May 12 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..246

Main diagonal of A363007.
Main diagonal of A153278 (for n>=1).
Cf. A139383, A261280, A346802, A351433.

b:= proc(n, t, m) option remember; `if`(n=0, `if`(t<2, m!,
      b(m, t-1, 0)), m*b(n-1, t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 12 2023

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

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A003659 Shifts left under Stirling2 transform.

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A363007 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - f^k(x)), where f(x) = exp(x) - 1.

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A363008 Expansion of e.g.f. 1/(2 - exp(exp(exp(exp(x) - 1) - 1) - 1)).

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A363009 Expansion of e.g.f. 1/(2 - exp(exp(exp(exp(exp(x) - 1) - 1) - 1) - 1)).

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A099391 Expansion of e.g.f. 1/(2 - exp(exp(exp(x) - 1) - 1)).

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A363010 a(n) = n! * [x^n] 1/(1 - f^n(x)), where f(x) = exp(x) - 1.

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