A290219 -id:A290219 - cp's OEIS frontend

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

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

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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).

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

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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

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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).

A330603 a(n) = Sum_{k>=0} (k - n)^n / 2^(k + 1).

Original entry on oeis.org

1, 0, 3, -14, 155, -1834, 27867, -492246, 10068459, -232990178, 6025718963, -172182404734, 5387697769467, -183214963001082, 6728091949444491, -265348057242998822, 11185888456798395563, -501937946696294628946, 23886968118494957119011, -1201674025637823778926414

Offset: 0

Ilya Gutkovskiy, Dec 19 2019

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The n-th term of the n-th inverse binomial transform of A000670.

G. C. Greubel, Table of n, a(n) for n = 0..380

Cf. A000670, A052841, A290219, A292916.

R:=PowerSeriesRing(Rationals(), 50);
A330603:= func< n | Coefficient(R!(Laplace( Exp(-n*x)/(2-Exp(x)) )), n) >;
[A330603(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

Table[Sum[(k - n)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 19}]
Table[HurwitzLerchPhi[1/2, -n, -n]/2, {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[-n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]

[factorial(n)*( exp(-n*x)/(2-exp(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

a(n) = n! * [x^n] exp(-n*x) / (2 - exp(x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-n)^(n - k) * A000670(k).
a(n) ~ (-1)^n * n^n / (2 - exp(-1)). - Vaclav Kotesovec, Dec 19 2019

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

A361380 Sum over the j-th term of the (n-j)-th inverse binomial transform of the Bell numbers (A000110) for all j in [n].

Original entry on oeis.org

1, 2, 3, 6, 17, 56, 215, 922, 4305, 21894, 119539, 696632, 4314925, 28237146, 194602079, 1407456694, 10649642837, 84100177424, 691474151187, 5907288773554, 52340230286509, 480153099982726, 4553711640946919, 44584683333637168, 450075389309517849

Offset: 0

Alois P. Heinz, Mar 09 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..576

Antidiagonal sums of A361781.

Cf. A000110, A059841, A290219, A347420.

a:= n-> add(add(binomial(i, j)*(i-n)^(i-j)*combinat[bell](j), j=0..i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> add(i!*coeff(series(exp(exp(x)-(n-i)*x-1), x, i+1), x, i), i=0..n):
seq(a(n), n=0..25);
# third 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-> add(b(i, i-n), i=0..n):
seq(a(n), n=0..25);

from math import comb
from sympy import bell
def A361380(n): return sum(comb(i,j)*(i-n)**(i-j)*bell(j) for i in range(n+1) for j in range(i+1)) # Chai Wah Wu, Apr 05 2023

a(n) = Sum_{i=0..n} i! * [x^i] exp(exp(x)-(n-i)*x-1).
a(n) = Sum_{0<=j<=i<=n} binomial(i,j)*(i-n)^(i-j)*Bell(j).
a(n) mod 2 = A059841(n).

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).

A340823 a(n) = exp(-1) * Sum_{k>=0} (k*(k - n))^n / k!.

Original entry on oeis.org

1, 1, 3, 5, 124, -2075, 91993, -4709903, 312334595, -25531783799, 2524083665172, -296260739274275, 40667620527027177, -6446882734412545043, 1167717545574222779643, -239452569059443831797303, 55146244227862697483251020, -14163492441645773105212592623

Offset: 0

Ilya Gutkovskiy, Jan 22 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..260

Cf. A000110, A020556, A020557, A290219, A334243, A340822.

A340823:= func< n | (&+[(-n)^j*Binomial(n,j)*Bell(2*n-j): j in [0..n]]) >;
[A340823(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

Table[Exp[-1] Sum[(k (k - n))^n/k!, {k, 0, Infinity}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] (-n)^k, {k, 0, n}], {n, 1, 17}]]

def A340823(n): return sum( binomial(n,k)*bell_number(2*n-k)*(-n)^k for k in range(n+1))
[A340823(n) for n in range(31)] # G. C. Greubel, Jun 12 2024

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * (-n)^k.

cp's OEIS Frontend

A346738 Expansion of e.g.f.: exp(exp(x) - 3*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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A330603 a(n) = Sum_{k>=0} (k - n)^n / 2^(k + 1).

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

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A361380 Sum over the j-th term of the (n-j)-th inverse binomial transform of the Bell numbers (A000110) for all j in [n].

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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.

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