A126617 -id:A126617 - cp's OEIS frontend

A126390 a(n) = Sum_{i=0..n} 2^iB(i)binomial(n,i) where B(n) = Bell numbers A000110(n).

1, 3, 13, 71, 457, 3355, 27509, 248127, 2434129, 25741939, 291397789, 3510328695, 44782460313, 602513988107, 8518757813637, 126179029108463, 1952609274344353, 31492811964616163, 528249539951292461, 9197240228562763687, 165923214676585626729

Offset: 0

N. J. A. Sloane, Aug 04 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.

Cf. A000110, A000296, A005493, A124311, A126617, A284859, A285064.

with(combstruct):seq(count(([S, {N=Union(Z, S, P), S=Set(Union(Z, P), card>=0), P=Set(Union(Z, Z), card>=1)}, labeled], size=n)), n=0..20); # Zerinvary Lajos, Mar 18 2008

Table[ Sum[ 2^k Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 30}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)

x='x+O('x^66); Vec(serlaplace((exp(exp(2*x)-1+x)))) \\ Joerg Arndt, May 13 2013

E.g.f.: exp(exp(2*x)-1+x). - Vladeta Jovovic, Aug 04 2007
a(n) = e^(-1)* 2^n * Sum_{k>=0} (k + 1/2)^n / k!. This is a Dobinski-type formula. - Karol A. Penson and Olivier Gérard, Oct 22 2007
G.f.: 1/Q(0), where Q(k)= 1 - (2*k+3)*x - 4*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x/(1 - 2*x*(2*k+1)/(1 - x - 2*x/(1 - 2*x*(2*k+2)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^k * a(n-k). - Ilya Gutkovskiy, Jun 21 2022
From Vaclav Kotesovec, Jun 22 2022: (Start)
a(n) ~ Bell(n) * (2 + LambertW(n)/n)^n.
a(n) ~ Bell(n) * 2^n * sqrt(n) * log(n)^(-1/2 + 1/(2*log(n)) - 1/(2*log(n)^2)) * exp(log(log(n))^2/(4*log(n)^2)). (End)
a(n) ~ 2^n * n^(n + 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2022

A124311 a(n) = Sum_{i=0..n} (-2)^ibinomial(n,i)B(i) where B(n) = Bell numbers A000110(n).

Original entry on oeis.org

1, -1, 5, -21, 121, -793, 5917, -49101, 447153, -4421105, 47062773, -535732805, 6484924585, -83079996041, 1121947980173, -15915567647101, 236442490569825, -3668776058118881, 59316847871113445, -997182232031471477, 17397298225094055897, -314449131128077197561

Offset: 0

N. J. A. Sloane, Aug 04 2007

sign

The sequence has strictly alternating signs. The variant Dobinski-type formula e^(-1)* (2)^n * Sum_{k >= 0} ( (k-1/2)^n / k! ) is strictly positive. - Karol A. Penson and Olivier Gérard, Oct 22 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000110, A000296, A005493, A126390, A126617.

A124311:= func< n | (&+[(-2)^k*Binomial(n,k)*Bell(k): k in [0..n]]) >;
[A124311(n): n in [0..30]]; // G. C. Greubel, Aug 25 2023

Table[ Sum[ (-2)^(k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)
With[{nn=30},CoefficientList[Series[Exp[Exp[-2x]-1+x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2016 *)

def A124311_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a + 2*(k*(b+c)+c)
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
        R.append((-1)^m*sum(T))
    return R
A124311_list(22)  # Peter Luschny, Nov 02 2012

def A124311(n): return sum( (-2)^k*binomial(n,k)*bell_number(k) for k in range(n+1) )
[A124311(n) for n in range(31)] # G. C. Greubel, Aug 25 2023

E.g.f.: exp(exp(-2*x) - 1 + x). - Vladeta Jovovic, Aug 04 2007
G.f.: 1/U(0) where U(k)=  1 + x*(2*k+1) - 4*x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
a(n) ~ (-2)^n * n^(n - 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * (-2)^k * a(n-k). - Ilya Gutkovskiy, Nov 29 2023

A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).

Original entry on oeis.org

1, -2, 5, -13, 36, -101, 293, -848, 2523, -7365, 22402, -64395, 205285, -541802, 2057617, -3403993, 28685420, 43885023, 824532745, 4878097904, 44263112047, 357891860463, 3169228222338, 28506399763969, 266822555964441, 2573194635922990, 25606751525353741

Offset: 0

Ilya Gutkovskiy, Jul 31 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..579

Cf. A000110, A000296, A005494, A126617, A193683, A290219, A346739, A346740.

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(Exp(x)-3*x-1) ))) // G. C. Greubel, Jun 12 2024

nmax = 26; CoefficientList[Series[Exp[Exp[x] - 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k], {k, 0, n}], {n, 0, 26}]
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]

[factorial(n)*( exp(exp(x)-3*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 + 3*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 3)^n / k!.
a(0) = 1; a(n) = -3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

Original entry on oeis.org

1, 0, 2, -13, 127, -1573, 23711, -421356, 8626668, -199971255, 5177291275, -148078588667, 4636966634653, -157786054331852, 5797411243015250, -228749440644895405, 9646951350227609155, -433035586385769361001, 20614401475233006857035, -1037331650810058231498688

Offset: 0

Ilya Gutkovskiy, Oct 06 2017

sign

The n-th term of the n-th inverse binomial transform of A000110.

Seiichi Manyama, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.

Main diagonal of A361781.

R:=PowerSeriesRing(Rationals(), 50);
A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;
[A290219(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-> b(n, -n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]

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

A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).

Original entry on oeis.org

1, -3, 10, -35, 127, -472, 1787, -6855, 26572, -103765, 407695, -1608378, 6369117, -25271183, 100542930, -400114103, 1597052419, -6359524256, 25481982047, -101103395443, 409291679676, -1592903606657, 6729506287091, -23748796926026, 123501587468073, -227183793907851

Offset: 0

Ilya Gutkovskiy, Jul 31 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..580

Cf. A000110, A000296, A045379, A126617, A193684, A290219, A346738, A346740.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(Exp(x) -4*x -1) ))) // G. C. Greubel, Jun 12 2024

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 4 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k], {k, 0, n}], {n, 0, 25}]
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

[factorial(n)*( exp(exp(x) -4*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 + 4*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 4)^n / k!.
a(0) = 1; a(n) = -4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

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).

A135494 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with lowering operator (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } where T(x) is Cayley's Tree function.

Original entry on oeis.org

1, -1, 1, -1, -3, 1, -1, -1, -6, 1, -1, 5, 5, -10, 1, -1, 19, 30, 25, -15, 1, -1, 49, 49, 70, 70, -21, 1, -1, 111, -70, -91, 70, 154, -28, 1, -1, 237, -883, -1218, -861, -126, 294, -36, 1, -1, 491, -4410, -4495, -3885, -2877, -840, 510, -45, 1

Offset: 1

Tom Copeland, Feb 08 2008

The lowering (or delta) operator for these polynomials is L = (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } and the raising operator is R = 2t * { 1 - T[ (1/2) * exp[(D-1)/2] ] }, where T(x) is the tree function of A000169. In addition, L = E(D,1) = A(D) where E(x,t) is the e.g.f. of A134991 and A(x) is the e.g.f. of A000311, so L = sum(j=1,...) A000311(j) * D^j / j! also. The polynomials and operators can be generalized through A134991.
Also the Bell transform of A153881. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
Exponential Riordan array [2 - exp(x), 1 + 2*x - exp(x)] belonging to the derivative subgroup of the exponential Riordan group. See the example section for a factorization of this array as an infinite product of arrays. - Peter Bala, Feb 13 2025

			The triangle begins:
  [1]  1;
  [2] -1,  1;
  [3] -1, -3,  1;
  [4] -1, -1, -6,   1;
  [5] -1,  5,  5, -10,   1;
  [6] -1, 19, 30,  25, -15,   1;
  [7] -1, 49, 49,  70,  70, -21, 1.
P(3,t) = [B(.,-t) + 2t]^3 = B(3,-t) + 3B(2,-t)2t + 3B(1,-t)(2t)^2 + (2t)^3 = (-t + 3t^2 - t^3) + 3(-t + t^2)(2t) + 3(-t)(2t)^2 + (2t)^3 = -t - 3t + t^3.
From _Peter Bala_, Feb 13 2025: (Start)
The array factorizes as an infinite product of lower triangular arrays:
  /  1               \    / 1             \ / 1             \ / 1             \
  | -1   1           |   | -1  1          | | 0 -1          | | 0  1          |
  | -1  -3   1       | = | -1 -2   1      | | 0 -1  1       | | 0  0  1       | ...
  | -1  -1  -6   1   |   | -1 -3  -3  1   | | 0 -1 -2  1    | | 0  0 -1  1    |
  | -1   5   5 -10  1|   | -1 -4  -6 -4  1| | 0 -1 -3 -3  1 | | 0  0 -1 -2  1 |
  |...               |   |...             | |...            | |...            |
where the first array in the product on the right-hand side is A154926. (End)

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.

Vincenzo Librandi, Rows n = 1..25
Peter Bala, Factorisations of some Riordan arrays as infinite products
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95.

Cf. A000169, A000311, A008277, A074051, A126617, A134991, A154926, A264428.

Cf. A298673 for the inverse matrix.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,1,-1), 9); # Peter Luschny, Jan 27 2016

max = 8; s = Series[Exp[t*(-Exp[x]+2*x+1)], {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]*n!; Table[t[n, k], {n, 0, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 23 2014 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 1, -1] &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Row polynomials are P(n,t) = Sum_{j=1..n} C(n,j) * t^j = [ Bell(.,-t) + 2t ]^n, umbrally, where Bell(j,t) are the Touchard/Bell/exponential polynomials described in A008277, with P(0,t) = 1.
E.g.f.: exp{ t * [ -exp(x) + 2x + 1] } and [ P(.,t) + P(.,s) ]^n = P(n,s+t).
The lowering operator gives L[P(n,t)] = n * P(n-1,t) = (D-1)/2 * P(n,t) + Sum_{j>=1} j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2).
The raising operator gives R[P(n,t)] = P(n+1,t) = 2t * { P(n,t) - Sum_{j>=1} j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2) } .
Therefore P(n+1,t) = 2t * { [ (1+D)/2 * P(n,t) ] - n * P(n-1,t) }.
P(n,1) = (-1)^n * A074051(n) and P(n,-1) = A126617(n).
See Rota, Roman, Mathworld or Wikipedia on Sheffer sequences and umbral calculus for more formulas, including expansion theorems.
From Tom Copeland, Jan 20 2018: (Start)
Define Q(n,z;w) = [Bell(.,w)+z]^n. Then Q(n,z;w) are a sequence of Appell polynomials with e.g.f. exp[(exp(t)-1+z)*w], lowering operator D = d/dz, and raising operator R = z + w*exp(D), and exp[(exp(D)-1)w] z^n = exp[Bell(.,w)D] z^n = Q(n,z;w) = e^(-w) (w d/dw + z)^n e^w =  e^(-w) exp(a.w) = exp[(a. - 1)w] with (a.)^k = a_k = (k + z)^n and (a. - 1)^m = sum{k = 0,..,m} (-1)^k a^(m-k). Then P(n,t) = Q(n,2t;-t).
For example, exp[(a. - 1)w] = (a. - 1)^0 + (a. - 1)^1 w + (a. - 1)^2 w^2/2! + ... = a_0 + (a_1 - a_0) w + (a_2 - 2a_1 + a_0) w^2/2! + ... = z^n + [(1+z)^n - z^n] w + [(2+z)^n - 2(1+z)^n + z^n] w^2/2! + ... . (End)
T(n+1, k) = Sum_{i = 0..n} s(n,k)*binomial(n, i)*T(i, k-1), where s(n,i) = 1 if i = n else -1. - Peter Bala, Feb 13 2025

More terms from Vincenzo Librandi, Jan 21 2018

A153732 Binomial transform of A109747.

Original entry on oeis.org

1, 3, 8, 19, 41, 84, 171, 347, 690, 1385, 2825, 5438, 11077, 24535, 33720, 102623, 350605, -1120228, 5876775, 11232063, -256532422, 1748895117, -4057110163, -42841409122, 605093026361, -3691581277925, 3538657621384, 186391745956155, -2296017574506751

Offset: 0

Gary W. Adamson, Dec 31 2008

sign

Equals triple binomial transform of A014182.

			a(3) = 19 = (1, 3, 3, 1) dot (1, 2, 3, 3) = (1 + 6 + 9 + 3); where A109747 = (1, 2, 3, 3, 2, 3, 5, -4, 5, 55, -212, ...).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A109747, A014182, A126617.

Join[{1}, Rest[CoefficientList[Series[Exp[2*x + 1 - Exp[-x]], {x, 0, 50}], x]*Range[0, 50]!]] (* G. C. Greubel, Aug 31 2016 *)

A007318 * A109747.
E.g.f.: exp(2*x+1-exp(-x)) = 1+3*x+8*x^2/2!+19*x^3/3!+....
a(n) = exp(1)*Sum_{k >= 0} (-1)^k*(2-k)^n/k!. Cf. A126617. - Peter Bala, Oct 28 2011.
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1+k*x-2*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
a(0) = 1; a(n) = 2*a(n-1) - Sum_{k=1..n} (-1)^k * binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 01 2023

A367888 Expansion of e.g.f. exp(3(exp(x) - 1) - 2x).

Original entry on oeis.org

1, 1, 4, 13, 61, 304, 1747, 10945, 74830, 550687, 4335109, 36272086, 320980645, 2991373597, 29253607780, 299258487553, 3193634980753, 35469069928792, 409082335024591, 4890313138089133, 60489400453642822, 772967507343358171, 10189818916331129017, 138398721137005215526

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A027710, A126617, A194689, A355254, A367889.

b:= proc(n, k, m) option remember; `if`(n=0, 3^m, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 29 2025

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-2)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A027710(k).

A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 5, 1, -2, 2, 1, 15, 1, -3, 5, -3, 4, 52, 1, -4, 10, -13, 7, 11, 203, 1, -5, 17, -35, 36, -10, 41, 877, 1, -6, 26, -75, 127, -101, 31, 162, 4140, 1, -7, 37, -139, 340, -472, 293, -21, 715, 21147, 1, -8, 50, -233, 759, -1573, 1787, -848, 204, 3425, 115975

Offset: 0

Alois P. Heinz, Mar 23 2023

			Square array A(n,k) begins:
    1,   1,   1,    1,     1,      1,       1,       1, ...
    1,   0,  -1,   -2,    -3,     -4,      -5,      -6, ...
    2,   1,   2,    5,    10,     17,      26,      37, ...
    5,   1,  -3,  -13,   -35,    -75,    -139,    -233, ...
   15,   4,   7,   36,   127,    340,     759,    1492, ...
   52,  11, -10, -101,  -472,  -1573,   -4214,   -9685, ...
  203,  41,  31,  293,  1787,   7393,   23711,   63581, ...
  877, 162, -21, -848, -6855, -35178, -134873, -421356, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-5 give: A000110, A000296, A126617, A346738, A346739, A346740.
Rows n=0-2 give: A000012, A024000, A160457.
Main diagonal gives A290219.
Antidiagonal sums give A361380.
Cf. A108087.

T:= func< n,k | (&+[(-k)^j*Binomial(n,j)*Bell(n-j): j in [0..n]]) >;
A361781:= func< n,k | T(k, n-k) >;
[A361781(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024

A:= proc(n, k) option remember; uses combinat;
      add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
A:= (n, k)-> b(n, -k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

T[n_, k_]:= T[n, k]= If[k==0, BellB[n], Sum[(-k)^j*Binomial[n,j]*BellB[n-j], {j,0,n}]];
A361781[n_, k_]= T[k, n-k];
Table[A361781[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 12 2024 *)

def T(n,k): return sum( (-k)^j*binomial(n,j)*bell_number(n-j) for j in range(n+1))
def A361781(n, k): return T(k, n-k)
flatten([[A361781(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

E.g.f. of column k: exp(exp(x) - k*x - 1).

A(n,k) = Sum_{j=0..n} (-k)^j*binomial(n,j)*Bell(n-j).

cp's OEIS Frontend

A126390 a(n) = Sum_{i=0..n} 2^i*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n).

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A124311 a(n) = Sum_{i=0..n} (-2)^i*binomial(n,i)*B(i) where B(n) = Bell numbers A000110(n).

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A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).

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A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

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A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).

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

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A135494 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with lowering operator (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } where T(x) is Cayley's Tree function.

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A126390 a(n) = Sum_{i=0..n} 2^iB(i)binomial(n,i) where B(n) = Bell numbers A000110(n).

A124311 a(n) = Sum_{i=0..n} (-2)^ibinomial(n,i)B(i) where B(n) = Bell numbers A000110(n).

A367888 Expansion of e.g.f. exp(3(exp(x) - 1) - 2x).