A196834 -id:A196834 - cp's OEIS frontend

A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).

1, 4, 17, 77, 372, 1915, 10481, 60814, 372939, 2409837, 16360786, 116393205, 865549453, 6713065156, 54190360453, 454442481041, 3952241526188, 35590085232519, 331362825860749, 3185554606447814, 31581598272055879, 322516283206446897, 3389017736055752178

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

For further information, references, programs, etc. for r-Bell numbers see A005493. - N. J. A. Sloane, Nov 27 2013
From expansion of falling factorials (binomial transform of A005493).
Row sums of Sheffer triangle (exp(3*x), exp(x)-1). - Wolfdieter Lang, Sep 29 2011

			G.f. = 1 + 4*x + 17*x^2 + 77*x^3 + 372*x^4 + 1915*x^5 + 10481*x^6 + 60814*x^7 + ...

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
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.
J. Riordan, Letter, Oct 31 1977
N. J. A. Sloane, Transforms
Earl Glen Whitehead Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.

Cf. A000110, A005493, A045379, A108087, A124323, A196834.

A row or column of the array A108087.

A005494:= func< n | (&+[Binomial(n,j)*3^(n-j)*Bell(j): j in [0..n]]) >;
[A005494(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

seq(add(3^(n-i)*combinat:-bell(i)*binomial(n,i),i=0..n), n=0..50); # Robert Israel, Dec 16 2014
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0,
      m^2, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n+1, 0)-b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2025

Range[0, 40]! CoefficientList[Series[Exp[3 x + Exp[x] - 1], {x, 0, 40}], x] (* Vincenzo Librandi, Mar 04 2014 *)

def A005494(n): return sum( 3^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A005494(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = Sum_{i=0..n} 3^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007
a(n) = exp(-1)*Sum_{k>=0} ((k+3)^n)/k!. - Gerald McGarvey, Jun 03 2004. May be rewritten as a(n) = Sum_{k>=3} (k^n*(k-1)*(k-2)/k!)/exp(1), which is a Dobinski-type relation for this sequence. - Karol A. Penson, Aug 18 2006
Define f_1(x), f_2(x), ... such that f_1(x) = x^2*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)charpoly(A,-3). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=3..n+3} A143495(n+3,k), n >= 0. - Wolfdieter Lang, Sep 29 2011
G.f.: 1/U(0) where U(k)= 1 - x*(k+4) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: Sum_{k>0} x^(k-1) / ((1 - 3*x) * (1 - 4*x) * ... * (1 - (k+2)*x)). - Michael Somos, Feb 26 2014
G.f.: Sum_{k>0} k * x^(k-1) / ((1 - 2*x) * (1 - 3*x) * ... * (1 - (k+1)*x)). - Michael Somos, Feb 26 2014
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 3) / LambertW(n)^(n + 7/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{k=0..n} 4^k*A124323(n, k). - Mélika Tebni, Jun 10 2022

A193685 5-Stirling numbers of the second kind.

Original entry on oeis.org

1, 5, 1, 25, 11, 1, 125, 91, 18, 1, 625, 671, 217, 26, 1, 3125, 4651, 2190, 425, 35, 1, 15625, 31031, 19981, 5590, 740, 45, 1, 78125, 201811, 170898, 64701, 12250, 1190, 56, 1, 390625, 1288991, 1398097, 688506, 174951, 24150, 1806, 68, 1, 1953125, 8124571, 11075670, 6906145, 2263065, 416451, 44016, 2622, 81, 1, 9765625, 50700551, 85654261, 66324830, 27273730, 6427575, 900627, 75480, 3675, 95, 1

Offset: 0

Wolfdieter Lang, Oct 06 2011

This is the lower triangular Sheffer matrix (exp(5*x),exp(x)-1). For Sheffer matrices see the W. Lang link under A006232 with references, and the rules for the conversion to the umbral notation of S. Roman's book.
The general case is Sheffer (exp(r*x),exp(x)-1), r=0,1,..., corresponding to r-Stirling2 numbers with row and column offsets 0. See the Broder link for r-Stirling2 numbers with offset [r,r].
a(n,m), n >= m >= 0, gives the number of partitions of the set {1.2....,n+5} into m+5 nonempty distinct subsets such that 1,2,3,4 and 5 belong to distinct subsets.
a(n,m) appears in the following normal ordering of Bose operators a and a* satisfying the Lie algebra [a,a*]=1: (a*a)^n (a*)^5 = Sum_{m=0..n} a(n,m)*(a*)^(5+m)*a^m, n >= 0. See the Mikhailov papers, where a(n,m) = S(n+5,m+5,5).
  With a->D=d/dx and a*->x we also have
  (xD)^n x^5 = Sum_{m=0..n} a(n,m)*x^(5+m)*D^m, n >= 0.

			n\m  0    1    2   3  4  5 ...
0    1
1    5    1
2   25   11    1
3  125   91   18   1
4  625  671  217  26  1
5 3125 4651 2190 425 35  1
...
5-restricted S2: a(1,0)=5 from 1,6|2|3|4|5, 2,6|1|3|4|5,
3,6|1|2|4|5, 4,6|1|2|3|5 and 5,6|1|2|3|4.
Recurrence: a(4,2) = (5+2)*a(3,2)+ a(3,1) = 7*18 + 91 = 217.
Normal ordering (n=1): (xD)^1 x^5 = Sum_{m=0..1} a(1,m)*x^(5+m)*D^m = 5*x^5 + 1*x^6*D.
a(2,1) = Sum_{j=0..1} S1(5,5-j)*S2(7-j,6) = 1*21 - 10*1 = 11.

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Generalized Dobinski formulas
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984)
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.

Cf. A048993, A143494, A143495, A143496.
Cf. A196834 (row sums), A196835 (alternating row sums).
Columns: A000351 (m=0), A005062 (m=1), A019757 (m=2), A028165 (m=3), ...

a[n_, m_] := Sum[ StirlingS1[5, 5-j]*StirlingS2[n+5-j, m+5], {j, 0, Min[5, n-m]}]; Flatten[ Table[ a[n, m], {n, 0, 10}, {m, 0, n}] ] (* Jean-François Alcover, Dec 02 2011, after Wolfdieter Lang *)

E.g.f. of row polynomials s(n,x):=Sum_{m=0..n} a(n,m)*x^m: exp(5*z + x(exp(z)-1)).
E.g.f. of column no. m (with leading zeros):
  exp(5*x)*((exp(x)-1)^m)/m!, m >= 0 (Sheffer).
O.g.f. of column no. m (without leading zeros):
  1/Product_{j=0..m} (1-(5+j)*x), m >= 0. (Compute the first derivative of the column e.g.f. and compare its Laplace transform with the partial fraction decomposition of the o.g.f. x^(m-1)/Product_{j=0..m} (1-(5+j)*x). This works for every r-restricted Stirling2 triangle.)
Recurrence: a(n,m) = (5+m)*a(n-1,m) + a(n-1,m-1), a(0,0)=1, a(n,m)=0 if n < m, a(n,-1)=0.
a(n,m) = Sum_{j=0..min(5,n-m)} S1(5,5-j)*S2(n+5-j,m+5), n >= m >= 0, with S1 and S2 the Stirling1 and Stirling2 numbers A008275 and A048993, respectively (see the Mikailov papers).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(4+k)^(n-1)*x^(k-1)/k!. - Peter Bala, Jun 23 2014

A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

Original entry on oeis.org

1, 5, 26, 141, 799, 4736, 29371, 190497, 1291020, 9131275, 67310847, 516369838, 4116416797, 34051164985, 291871399682, 2588914083065, 23733360653955, 224592570163192, 2191466128865567, 22024934452712437, 227771488390279260

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Cf. A000110, A005493, A005494, A108087, A196834.

Equals the row sums of triangle A143496. - Wolfdieter Lang, Sep 29 2011

A045379:= func< n | (&+[Binomial(n,j)*4^(n-j)*Bell(j): j in [0..n]]) >;
[A045379(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

a[0]= 1; a[n_]:= a[n]= 4*a[n-1] +Sum[Binomial[n-1, k]*a[k], {k,0,n-1}]; Array[a, 21, 0] (* Amiram Eldar, Jul 03 2020 *)

def A045379(n): return sum( 4^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A045379(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = exp(-1)*Sum_{k>=0} ((k+4)^n)/k!. - Gerald McGarvey, Jun 03 2004
A recursive formula to compute some integer sequences (including A000110, A005493, A005494 and the present sequence). Define G(n, m), where n, m >= 0, as follows: G(0, m) = 1; G(n, m) = G(n-1, m) * (m+1) + G(n-1, m+1), where n > 0. Then G(n, 0) = A000110(n+1); G(n, 1) = A005493(n+1); G(n, 2) = A005494(n+1); G(n, 3) = A045379(n+1). - Alexey Andreev (ava12(AT)nm.ru), Jan 05 2006
Define f_1(x), f_2(x), ... such that f_1(x)=x^3*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)*charpoly(A,-4). - Milan Janjic, Jul 08 2010
G.f.: 1/U(0) where U(k) = 1 - x*(k+5) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 4) / LambertW(n)^(n + 9/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{j=0..n} binomial(n, j)*4^(n-j)*A000110(j). - G. C. Greubel, Dec 01 2022

A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

Original entry on oeis.org

1, 4, 15, 51, 146, 273, -319, -6374, -36235, -113833, 69388, 3772035, 28631669, 112704452, -96418909, -5652669753, -50538496446, -230554460867, 281597003109, 16303457144146, 166512491229617, 872578914956059, -1111135578108284, -78512971676777833, -919653124088665479

Offset: 0

Wolfdieter Lang, Oct 07 2011

			a(2) = 25 - 11 + 1 = 15.

Seiichi Manyama, Table of n, a(n) for n = 0..589

Cf. A193685, A196834 (row sums), A193683, A193684, A293037, A346740.

my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+5*x+1))) \\ Michel Marcus, Aug 02 2021

a(n) = Sum_{m=0..n} (-1)^m * A193685(n,m), n>=0.
E.g.f.: exp(-exp(x)+5*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 5)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 5 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).

Original entry on oeis.org

1, -4, 17, -75, 340, -1573, 7393, -35178, 169035, -818603, 3989250, -19538555, 96084397, -474052868, 2344993157, -11624422855, 57722000172, -287012948441, 1428705217949, -7118044107698, 35489117143047, -177036294035559, 883588566571138, -4411213326568599, 22032317835916969

Offset: 0

Ilya Gutkovskiy, Jul 31 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..581

Cf. A000110, A000296, A126617, A196834, A196835, A290219, A346738, A346739.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(Exp(x) -5*x -1) ))) // G. C. Greubel, Jun 12 2024

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 5 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-5)^(n - k) BellB[k], {k, 0, n}], {n, 0, 24}]
a[0] = 1; a[n_] := a[n] = -5 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]

[factorial(n)*( exp(exp(x) -5*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 5*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-5)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 5)^n / k!.
a(0) = 1; a(n) = -5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 15, 5, 9, 13, 20, 17, 27, 37, 52, 6, 11, 16, 25, 21, 34, 47, 67, 26, 43, 60, 87, 77, 114, 151, 203, 7, 13, 19, 30, 25, 41, 57, 82, 31, 52, 73, 107, 94, 141, 188, 255, 37, 63, 89, 132, 115, 175, 235, 322, 141, 218, 295, 409, 372, 523, 674

Offset: 0

Mikhail Kurkov, Aug 23 2021 [verification needed]

Modulo 2 binomial transform of A243499(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Kevin Ryde, PARI/GP Code

Cf. A000110, A000120, A002720, A005493, A005494, A007814, A008277, A035009, A045379, A053645, A129760, A143494, A143495, A196834, A265649, A295989.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

function a = A347204(max_n)
    a(1) = 1;
    a(2) = 2;
    for nloop = 3:max_n
        n = nloop-1;
        s = 0;
        for k = 0:floor(log2(n))-1
            s = s + a(1+A053645(n)-2^k*(mod(floor(n/(2^k)),2)));
        end
        a(nloop) = 2*a(A053645(n)+1) + s;
    end
end
function a_n = A053645(n)
    a_n = n - 2^floor(log2(n));
end % Thomas Scheuerle, Oct 25 2021

f[n_] := BitAnd[n, n - 1]; a[0] = 1; a[n_] := a[n] = a[f[n]/2] + a[Floor[(n + f[n])/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 19 2021 *)

f(n) = bitand(n, n-1); \\ A129760
a(n) = if (n<=1, n+1, if (n%2, a(n\2)+a(n-1), a(f(n/2)) + a(n/2+f(n/2)))); \\ Michel Marcus, Oct 25 2021

PARI
```
\\ Also see links.
```

A129760(n) = bitand(n, n-1);
memoA347204 = Map();
A347204(n) = if (n<=1, n+1, my(v); if(mapisdefined(memoA347204,n,&v), v, v = if(n%2, A347204(n\2)+A347204(n-1), A347204(A129760(n/2)) + A347204(n/2+A129760(n/2))); mapput(memoA347204,n,v); (v))); \\ (Memoized version of Michel Marcus's program given above) - Antti Karttunen, Nov 20 2021

a(n) = a(n - 2^f(n)) + (1 + f(n))*a((n - 2^f(n))/2) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(2n+1) = a(n) + a(2n) for n >= 0.
a(2n) = a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) - 2^k*T(n,k)) for n > 1 with a(0) = 1, a(1) = 2, and where f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2.
Sum_{k=0..2^n - 1} a(k) = A035009(n+1) for n >= 0.
a((4^n - 1)/3) = A002720(n) for n >= 0.
a(2^n - 1) = A000110(n+1),
a(2*(2^n - 1)) = A005493(n),
a(2^2*(2^n - 1)) = A005494(n),
a(2^3*(2^n - 1)) = A045379(n),
a(2^4*(2^n - 1)) = A196834(n),
a(2^m*(2^n-1)) = T(n,m+1) is the n-th (m+1)-Bell number for n >= 0, m >= 0 where T(n,m) = m*T(n-1,m) + Sum_{k=0..n-1} binomial(n-1,k)*T(k,m) with T(0,m) = 1.
a(n) = Sum_{j=0..2^A000120(n)-1} A243499(A295989(n,j)) for n >= 0. Also A243499(n) = Sum_{j=0..2^f(n)-1} (-1)^(f(n)-f(j)) a(A295989(n,j)) for n >= 0 where f(n) = A000120(n). In other words, a(n) = Sum_{j=0..n} (binomial(n,j) mod 2)*A243499(j) and A243499(n) = Sum_{j=0..n} (-1)^(f(n)-f(j))*(binomial(n,j) mod 2)*a(j) for n >= 0 where f(n) = A000120(n).
Generalization:
b(n, x) = (1/x)*b((n - 2^f(n))/2, x) + (-1)^n*b(floor((2n - 2^f(n))/2), x) for n > 0 with b(0, x) = 1 where f(n) = A007814(n).
Sum_{k=0..2^n - 1} b(k, x) = (1/x)^n for n >= 0.
b((4^n - 1)/3, x) = (1/x)^n*n!*L_{n}(x) for n >= 0 where L_{n}(x) is the n-th Laguerre polynomial.
b((8^n - 1)/7, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A265649(n, k) for n >= 0.
b(2^n - 1, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A008277(n+1, k+1),
b(2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143494(n+2, k+2),
b(2^2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143495(n+3, k+3),
b(2^m*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*T(n+m+1, k+m+1, m+1) for n >= 0, m >= 0 where T(n,k,m) is m-Stirling numbers of the second kind.

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A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).

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A193685 5-Stirling numbers of the second kind.

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A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

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A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

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A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).

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A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

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