A196835 -id:A196835 - cp's OEIS frontend

A293037 E.g.f.: exp(1 + x - exp(x)).

1, 0, -1, -1, 2, 9, 9, -50, -267, -413, 2180, 17731, 50533, -110176, -1966797, -9938669, -8638718, 278475061, 2540956509, 9816860358, -27172288399, -725503033401, -5592543175252, -15823587507881, 168392610536153, 2848115497132448, 20819319685262839

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..593

Column k=1 of A293051.
Column k=1 of A335977.
Cf. A000587 (k=0), this sequence (k=1), A293038 (k=2), A293039 (k=3), A293040 (k=4).
Cf. A000296, A014182, A193683, A193684, A196835.

f:= series(exp(1 + x - exp(x)), x= 0, 101): seq(factorial(n) * coeff(f, x, n), n = 0..30); # Muniru A Asiru, Oct 31 2017
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n+1, 1):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 01 2021

m = 26; Range[0, m]! * CoefficientList[Series[Exp[1 + x - Exp[x]], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -1], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-exp(x)+1+x)))

a(n) = if(n==0, 1, -sum(k=0, n-2, binomial(n-1, k)*a(k))); \\ Seiichi Manyama, Aug 02 2021

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(k + 1)^n/k!. - Ilya Gutkovskiy, Jun 13 2019
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -1). - Vaclav Kotesovec, Jul 06 2020
a(0) = 1; a(n) = - Sum_{k=0..n-2} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

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

A193683 Alternating row sums of Sheffer triangle A143495 (3-restricted Stirling2 numbers).

Original entry on oeis.org

1, 2, 3, 1, -14, -59, -99, 288, 2885, 10365, 1700, -226313, -1535203, -4258630, 17243695, 284513877, 1688253890, 2750940953, -51540956455, -624352447488, -3470378651847, -496964048927, 204678286709292, 2311290490508227, 12611758414937801

Offset: 0

Wolfdieter Lang, Oct 06 2011

In order to have a lower triangular Sheffer matrix for A143495 one uses row and column offsets 0 (not 3).

			Row no. 3 of A143495 with [0,0] offset is [27,37,12,1], hence a(3)=27-37+12-1=1.

See A143495.

Seiichi Manyama, Table of n, a(n) for n = 0..591

Cf. A143495, A074051 (2-restricted Stirling2 case), A193684, A196835, A293037, A346738.

With[{nn=30},CoefficientList[Series[Exp[3x+1-Exp[x]],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2013 *)

E.g.f.: exp(-exp(x)+3*x+1).
G.f.: (1 - 2/E(0))/x where E(k) = 1 + 1/(1 - 2*x/(1 - 2*(k+1)*x/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: 1/U(0) where U(k) = 1 - x*(k+2) + x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: (1 - G(0) )/(x+1) where G(k) =  1 - 1/(1-k*x-3*x)/(1-x/(x+1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
G.f.: 1/Q(0), where Q(k) = 1 - 3*x + x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: G(0)/(1-2*x), where G(k) = 1 - x^2*(2*k+1)/(x^2*(2*k+1) + (1-x*(2*k+2))*(1-x*(2*k+3))/(1 - x^2*(2*k+2)/(x^2*(2*k+2) + (1-x*(2*k+3))*(1-x*(2*k+4))/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2014
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 3)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 3 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A193684 Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).

Original entry on oeis.org

1, 3, 8, 17, 17, -78, -585, -2021, -1710, 29395, 231413, 856264, -346979, -30019585, -232782792, -834712259, 2313820717, 59793779314, 469729578123, 1597321309383, -9914171906614, -206169178856073, -1697255630380351, -5677886943413120, 55801423903125353

Offset: 0

Wolfdieter Lang, Oct 06 2011

In order to have A143496 as a lower triangular Sheffer matrix one uses row and column offsets 0 (not 4).

			With offset [0,0] row n=3 of A143496 is [64,61,15,1], hence a(3)=64-61+15-1=17.

Seiichi Manyama, Table of n, a(n) for n = 0..591

Cf. A143496, A193683 (3-restricted Stirling2 case), A196835, A293037, A346739.

my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+4*x+1))) \\ Michel Marcus, Aug 02 2021

E.g.f.: exp(-exp(x)+4*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 4)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 4 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

Original entry on oeis.org

1, 6, 37, 235, 1540, 10427, 73013, 529032, 3967195, 30785747, 247126450, 2050937445, 17585497797, 155666739742, 1421428484337, 13377704321695, 129659127547372, 1293095848212799, 13259069937250169, 139671750579429512, 1510382932875294447, 16754464511605466311

Offset: 0

Wolfdieter Lang, Oct 07 2011

			a(2) = 25 + 11 + 1 = 37.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000110, A005493, A005494, A045379, A196835 (alternating row sums).

b:= proc(n, m) option remember;
     `if`(n=0, 1, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 5):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 22 2021

nmax = 20; CoefficientList[Series[E^(E^x + 5*x - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 10 2020 *)

a(n) = Sum_{m=0..n} A193685(n,m).
E.g.f.: exp(exp(x)+5*x-1).
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 5) / LambertW(n)^(n + 11/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 03 2020

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).

A298373 a(n) = n! * [x^n] exp(n*x - exp(x) + 1).

Original entry on oeis.org

1, 0, 0, 1, 17, 273, 4779, 93532, 2047730, 49854795, 1339872113, 39462731031, 1265248227869, 43895994373580, 1639148060192408, 65568985769784897, 2797922570156143597, 126880981472647625557, 6094210606862471240855, 309087628703330034215088, 16508178701980033054460042

Offset: 0

Ilya Gutkovskiy, Jan 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000587, A074051, A134980, A193683, A193684, A196835, A290219.

R:=PowerSeriesRing(Rationals(), 50);
A298373:= func< n | Coefficient(R!(Laplace( Exp(-Exp(x)+n*x+1) )), n) >;
[A298373(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k-1))
    end:
a:= n-> abs(b(n, -n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Table[n! SeriesCoefficient[Exp[n x - Exp[x] + 1], {x,0,n}], {n,0,20}]
Join[{1}, Table[Sum[Binomial[n, k] n^(n-k) BellB[k,-1] , {k,0,n}], {n,20}]]

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

a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000587(k).

a(n) ~ exp(1-exp(1)) * n^n. - Vaclav Kotesovec, Aug 04 2021

cp's OEIS Frontend

A293037 E.g.f.: exp(1 + x - exp(x)).

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A193685 5-Stirling numbers of the second kind.

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A193683 Alternating row sums of Sheffer triangle A143495 (3-restricted Stirling2 numbers).

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A193684 Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).

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A196834 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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A298373 a(n) = n! * [x^n] exp(n*x - exp(x) + 1).

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