A001861 -id:A001861 - cp's OEIS frontend

A194689 a(n) = Sum_{k=0..n} binomial(n,k)w(k)w(n-k) where w() = A000296().

1, 0, 2, 2, 14, 42, 222, 1066, 6078, 36490, 238046, 1653610, 12214270, 95361866, 784071966, 6764984362, 61066919230, 575200190986, 5640081557598, 57450510336234, 606773139773054, 6633515763375306, 74950634205257630, 873995513192234410, 10504736507220958142, 129983468625156713354

Offset: 0

N. J. A. Sloane, Sep 01 2011

nonn

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 771, Problem 37).

Seiichi Manyama, Table of n, a(n) for n = 0..560

Cf. A000296, A001861, A194689, A339014, A339017, A339027.

Table[Sum[(-1)^(n-k) * Binomial[n,k] * BellB[k,2] * 2^(n-k), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 25 2022 *)

N=66;  x='x+O('x^N);
Q(k) = if (k>N, 1, 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ) );
gf=1/Q(0);  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1-x)))) \\ Seiichi Manyama, Nov 20 2020

G.f.: 1/Q(0) where Q(k) = 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
G.f.: 1/Q(0), where Q(k)= 1 - x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 06 2013
E.g.f.: exp(2*(exp(x) - 1 - x)). - Ilya Gutkovskiy, Apr 07 2018
a(0) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-1-k). - Seiichi Manyama, Nov 20 2020
a(n) ~ 4 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). - Vaclav Kotesovec, Jun 26 2022

A000558 Generalized Stirling numbers of second kind.

Original entry on oeis.org

1, 6, 32, 175, 1012, 6230, 40819, 283944, 2090424, 16235417, 132609666, 1135846062, 10175352709, 95108406130, 925496853980, 9357279554071, 98118527430960, 1065259283215810, 11956366813630835, 138539436100687988, 1655071323662574756, 20361556640795422729

Offset: 2

N. J. A. Sloane

From Olivier Gérard, Mar 25 2009: (Start)
a(n) is the number of hierarchical partitions of a set of n elements into two second level classes : k>1 subsets of [n] are further grouped in two classes.
a(n) is equivalently the number of trees of uniform height 3 with n labeled leaves, and a root of order two. (End)

			From _Olivier Gérard_, Mar 25 2009: (Start)
a(2) = 1, since there is only one partition of {1,2} into two classes, and only one way to partition those classes.
a(4) = 32 = 7*1 + 6*3 + 1*7 since there are 7 ways of partitioning {1,2,3,4} into two classes (which cannot be grouped further), 6 ways of partitioning a set of 4 elements into three classes and three ways to partition three classes into two super-classes, etc. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..100
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
R. Fray, A generating function associated with the generalized Stirling numbers, Fib. Quart. 5 (1967), 356-366.

Column k=2 of A130191.
Cf. A000110, A000559, A046817.
Cf. A001861 for the related bicolor set partitions. - Olivier Gérard, Mar 25 2009

nn = 22; t = Range[0, nn]! CoefficientList[Series[1/2*(Exp[Exp[x] - 1] - 1)^2, {x, 0, nn}], x]; Drop[t, 2] (* T. D. Noe, Aug 10 2012 *)
a[n_] := Sum[StirlingS2[n, k] (2^(k-1)-1), {k, 0, n}];
a /@ Range[2, 100] (* Jean-François Alcover, Mar 30 2021 *)

E.g.f.: (1/2) * (exp(exp(x) - 1) - 1)^2. - Vladeta Jovovic, Sep 28 2003
a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(k,2). - Olivier Gérard, Mar 25 2009
a(n) = Sum_{k=1..n-1} binomial(n-1,k) * Bell(k) * Bell(n-k). - Ilya Gutkovskiy, Feb 15 2021

More terms from David W. Wilson, Jan 13 2000

A221159 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(3i).

Original entry on oeis.org

1, 8, 72, 712, 7624, 87496, 1067976, 13781448, 187104200, 2661876168, 39549629384, 611918940616, 9834596715464, 163824830616008, 2823080829871048, 50238768569014728, 921839901090823112, 17416746966515278280, 338394913332895863752, 6753431112631087835592, 138296031340416209103816

Offset: 0

N. J. A. Sloane, Jan 04 2013

nonn

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 8 labeled boxes. - Peter Bala, Mar 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1.

Cf. A000110, A001861, A078944, A221176, A027710, A144180, A144223, A144263, A189233.

Table[BellB[n,8],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

E.g.f.: exp(8*(exp(x) - 1)). - Peter Bala, Mar 23 2013
a(n) ~ n^n * exp(n/LambertW(n/8)-8-n) / (sqrt(1+LambertW(n/8)) * LambertW(n/8)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 8^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 11 2019

A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 4, 1, 52, 52, 37, 17, 5, 1, 203, 203, 151, 77, 26, 6, 1, 877, 877, 674, 372, 141, 37, 7, 1, 4140, 4140, 3263, 1915, 799, 235, 50, 8, 1, 21147, 21147, 17007, 10481, 4736, 1540, 365, 65, 9, 1, 115975, 115975, 94828, 60814, 29371, 10427, 2727, 537, 82, 10, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1).). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492.
Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - Gerald McGarvey, Oct 08 2006
Equals antidiagonals of an array in which (n+1)-th column is the binomial transform of n-th column, with leftmost column = the Bell sequence, A000110. - Gary W. Adamson, Apr 16 2009
Number of partitions of [n+k] where at least k blocks contain their own index element.  A(2,2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022

			Array A(n,k) begins:
   1,   1,   1,    1,    1,     1,     1,     1,     1,      1, ... A000012;
   1,   2,   3,    4,    5,     6,     7,     8,     9,     10, ... A000027;
   2,   5,  10,   17,   26,    37,    50,    65,    82,    101, ... A002522;
   5,  15,  37,   77,  141,   235,   365,   537,   757,   1031, ... A005491;
  15,  52, 151,  372,  799,  1540,  2727,  4516,  7087,  10644, ... A005492;
  52, 203, 674, 1915, 4736, 10427, 20878, 38699, 67340, 111211, ... ;
Antidiagonal triangle, T(n, k), begins as:
     1;
     1,    1;
     2,    2,    1;
     5,    5,    3,    1;
    15,   15,   10,    4,   1;
    52,   52,   37,   17,   5,   1;
   203,  203,  151,   77,  26,   6,  1;
   877,  877,  674,  372, 141,  37,  7,  1;
  4140, 4140, 3263, 1915, 799, 235, 50,  8,  1;

F. Ruskey, Combinatorial Generation, preprint, 2001.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1, Figure 1.
J. Riordan, Letter, Oct 31 1977, The array is on the second page.
F. Ruskey, Combinatorial Generation, 2003.
F. Ruskey, Lexicographic Algorithms [Broken link]

Cf. A000012, A000027, A000110, A002522, A005490, A005491.
Cf. A005492, A005493, A005494, A045379, A086659, A124824.
Main diagonal gives A134980.
Antidiagonal sums give A347420.

A108087:= func< n,k | (&+[Binomial(n-k,j)*k^j*Bell(n-k-j): j in [0..n-k]]) >;
[A108087(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 02 2022

with(combinat):
A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 18 2012

Unprotect[Power]; 0^0 = 1; A[n_, k_] := Sum[Binomial[n, i] * k^i * BellB[n - i], {i, 0, n}]; Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

f(n,k)=round (suminf(i=0,(i+k)^n/i!)/exp(1));
g(n,k)=for(k=0,k,print1(f(n,k),",")) \\ prints k+1 terms of n-th row

def A108087(n,k): return sum( k^j*bell_number(n-k-j)*binomial(n-k,j) for j in range(n-k+1))
flatten([[A108087(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022

For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)
A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
A(n,k) = Sum_{i=0..n} C(n,i) * k^i * Bell(n-i). - Alois P. Heinz, Jul 18 2012
Sum_{k=0..n-1} A(n-k,k) = A005490(n). - Alois P. Heinz, Jan 05 2022
From G. C. Greubel, Dec 02 2022: (Start)
T(n, n) = A000012(n).
T(n, n-1) = A000027(n).
T(n, n-2) = A002522(n-1).
T(n, n-3) = A005491(n-2).
T(n, n-4) = A005492(n+1).
T(2*n, n) = A134980(n).
T(2*n, n+1) = A124824(n), n >= 1.
Sum_{k=0..n} T(n, k) = A347420(n). (End)

A213170 E.g.f.: exp(2*(1-exp(x))).

Original entry on oeis.org

1, -2, 2, 2, -6, -14, 26, 178, 90, -2382, -9446, 13746, 287194, 998578, -3687782, -56264782, -208446118, 1017677490, 17194912282, 79540574642, -317691584294, -7577787031374, -47958156443238, 77252406086578, 4400217218583642, 39757699729476274, 54538870133137690

Offset: 0

Philippe Deléham, Feb 14 2013

Exponential self-convolution of complementary Bell numbers (A000587). - Vladimir Reshetnikov, Oct 07 2016

Seiichi Manyama, Table of n, a(n) for n = 0..580
Vaclav Kotesovec, Graph - asymptotic (1000 terms)

Cf. A000587, A001861, A007318, A048993.

b:= proc(n, m) option remember; `if`(n=0,
      (-2)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 17 2022

CoefficientList[Series[E^(2*(1-E^x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
Table[BellB[n, -2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

x='x+O('x^50); Vec(serlaplace(exp(2*(1-exp(x))))) \\ G. C. Greubel, Nov 15 2017

a(n) = Sum_{k=0..n} A048993(n,k)*(-2)^k.
a(n) = Sum_{k=0..n} A000587(k)*A000587(n-k)*binomial(n,k).
G.f.: 1/(1+2*x/(1-x/(1+2*x/(1-2*x/(1+2*x/(1-3*x/(1+2*x/(1-4*x/(1+2*x/(1-...(continued fraction).
Sum_{k=0..n} binomial(n,k)*a(k) = a(n+1)/(-2). - Philippe Deléham, Feb 17 2013
G.f.: 1/Q(0) where Q(k) = 1 + x*(k+1) + x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
Lim sup n->infinity (abs(a(n))/n!)^(1/n) / abs(exp(1/LambertW(-n/2)) / LambertW(-n/2)) = 1. - Vaclav Kotesovec, Aug 04 2014
a(n) = B_n(-2), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
G.f.: Sum_{j>=0} (-2)^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019

A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0

Offset: 0

Seiichi Manyama, Sep 25 2017

			Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,   1,    2,     3,     4,      5,      6, ...
   0,   2,    6,    12,    20,     30,     42, ...
   0,   5,   22,    57,   116,    205,    330, ...
   0,  15,   94,   309,   756,   1555,   2850, ...
   0,  52,  454,  1866,  5428,  12880,  26682, ...
   0, 203, 2430, 12351, 42356, 115155, 268098, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
Rows n=0..2 give A000012, A001477, A002378.
Main diagonal gives A242817.
Same array, different indexing is A189233.
Cf. A292861.

A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[, ] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, k). - Peter Luschny, Dec 23 2021

A002875 Sorting numbers (see Motzkin article for details).

Original entry on oeis.org

1, 2, 4, 24, 128, 880, 7440

Offset: 0

N. J. A. Sloane

How is the sequence defined (see the links in A000262)? Also more terms would be welcome.

Based on the Motzkin article, where this sequence appears in the last row of the table on p. 173, one would expect that this sequence is the same as A294202. However, they seem to be unrelated. So the true definition of this sequence is a mystery. - Andrew Howroyd and Andrey Zabolotskiy, Oct 25 2017

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A000262, A001861, A002872, A002873, A002874, A036073, A294201, A294202.

A108400 a(n) = Product_{k = 0..n} (2^k * k!).

Original entry on oeis.org

1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000

Offset: 0

Philippe Deléham, Jul 02 2005

Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886.
Hankel transform of the sequence with e.g.f. exp(x^2). Also (-1)^C(n+1,2)*a(n) is the Hankel transform of the sequence with e.g.f. exp(-x^2). - Paul Barry, Feb 12 2008
Let T(n,k) = (n+1)^k * (1+(-1)^(n-k))/2, then a(n) = det(T(i,j); 0<=i, j<=n). - Paul Barry, Feb 12 2008

G. C. Greubel, Table of n, a(n) for n = 0..38
M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A000178, A006125, A074962.

Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886.

BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >;
[2^Binomial(n+1,2)*BarnesG(n+2): n in [0..15]]; // G. C. Greubel, Jun 21 2022

Table[Product[k!*2^k, {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[2^Binomial[n+1,2]*BarnesG[n+2], {n,0,15}] (* G. C. Greubel, Jun 21 2022 *)

def barnes_g(n): return product(factorial(j) for j in (0..n-2))
[2^binomial(n+1,2)*barnes_g(n+2) for n in (0..15)] # G. C. Greubel, Jun 21 2022

a(n) = A006125(n+1)*A000178(n).
a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - Paul Barry, Aug 02 2008
a(n) ~ 2^((n+1)^2/2) * n^(n^2/2+n+5/12) * Pi^((n+1)/2) / (A * exp(3*n^2/4+n-1/12)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A036074 Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.

Original entry on oeis.org

1, 2, 9, 55, 412, 3619, 36333, 408888, 5080907, 68914023, 1011165446, 15935379409, 268125052373, 4792458452162, 90605469012877, 1805135197261131, 37775862401203916, 827992670793489263

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036076, ...

mx = 16; p = 4; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 4^k * BellB[k, 1/4] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n):=sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n); /* Vladimir Kruchinin, Sep 14 2010 */

a(n) = sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n), n > 0. - Vladimir Kruchinin, Sep 14 2010
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=4. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (4*n/LambertW(4*n))^n * exp(n/LambertW(4*n) + (4*n/LambertW(4*n))^(1/4) - n - 5/4) / sqrt(1 + LambertW(4*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters

A221176 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).

Original entry on oeis.org

1, 16, 272, 4880, 91920, 1810192, 37142288, 791744272, 17490370320, 399558315792, 9421351690000, 228916588400400, 5723078052339472, 147025755978698512, 3876566243300318992, 104789417805394595600, 2901159958960121863952, 82188946843192555474704, 2380551266738846355103504, 70441182699006212824911632

Offset: 0

N. J. A. Sloane, Jan 04 2013

nonn

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 16 labeled boxes. - Peter Bala, Mar 23 2013

Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1.

Cf. A000110, A001861, A078944, A221159. A027710, A144180, A144223, A144263, A189233.

With[{nn=20},CoefficientList[Series[Exp[16 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 19 2024 *)

E.g.f. exp(16*(exp(x) - 1)). - Peter Bala, Mar 23 2013

cp's OEIS Frontend

A194689 a(n) = Sum_{k=0..n} binomial(n,k)*w(k)*w(n-k) where w() = A000296().

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A000558 Generalized Stirling numbers of second kind.

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A221159 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(3i).

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A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

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A213170 E.g.f.: exp(2*(1-exp(x))).

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A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

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A002875 Sorting numbers (see Motzkin article for details).

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A194689 a(n) = Sum_{k=0..n} binomial(n,k)w(k)w(n-k) where w() = A000296().