A014182 -id:A014182 - cp's OEIS frontend

A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).

1, 1, -1, 1, 25, -674, 15211, -331827, 5987745, 15901597, -13125035449, 1292056076070, -103145930581319, 7462324963409941, -464957409070517453, 16313974895147212801, 2059903411953959582849, -708700955022151333496910, 143215213612865558214820303, -24681846509158429152517973103

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..282
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A009235, A014182, A292866, A301419, A317996, A318179, A318180, A318181.

Table[SeriesCoefficient[Sum[x^k/Product[(1 + n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[(-n)^(n - k) StirlingS2[n, k], {k, n}], {n, 19}]]
Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* Peter Luschny, Aug 20 2018 *)

{a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 27 2019

a(n) = n! * [x^n] exp((1 - exp(-n*x))/n), for n > 0.
a(n) = Sum_{k=0..n} (-n)^(n-k)*Stirling2(n,k).
a(n) = (-n)^n*BellPolynomial_n(-1/n) for n >= 1. - Peter Luschny, Aug 20 2018

A025016 Final digits of !n = Sum_{i=0..n} i! (A003422) for very large n, read from right.

Original entry on oeis.org

4, 1, 3, 0, 4, 9, 0, 2, 4, 0, 2, 9, 8, 2, 5, 6, 3, 3, 2, 4, 4, 6, 5, 5, 2, 5, 0, 9, 3, 0, 5, 0, 1, 3, 9, 5, 3, 2, 3, 4, 0, 8, 4, 9, 9, 7, 0, 1, 1, 2, 6, 8, 3, 7, 4, 8, 6, 8, 7, 4, 9, 7, 4, 7, 4, 2, 2, 9, 0, 0, 4, 3, 3, 0, 5, 6, 5, 8, 6, 5, 0, 0, 2, 6, 6, 5, 1, 5, 9, 7, 8, 8, 1, 6, 2, 0, 2, 8, 1, 2, 1, 3, 7, 6, 1, 1, 5, 8

Offset: 0

David W. Wilson

Reversed digits of 10-adic sum of all factorials.

More generally, the 10-adic sum: B(n) = Sum_{k>=0} k^n*k! is given by: B(n) = A014182(n)*B(0) + A014619(n) for n>=0, where B(0) is the 10-adic sum of factorials (this constant). - Paul D. Hanna, Aug 12 2006

			!20 = 256132749111820314, !30 = 16158688114800553828940314 ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A000142, A003422, A014182, A014619, A065356.

a[n_] := Module[{x, f=1}, While[Mod[f!, 10^(n+1)]>0, f += 1]; x = Sum[ Mod[k!, 10^(n+1)], {k, 0, f}]; Quotient[10*Mod[x, 10^(n+1)], 10^(n+1)]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 18 2015, after Paul D. Hanna *)

{a(n)=local(x,f=1);while(f!%10^(n+1)>0,f+=1); x=sum(k=0,f,k!%10^(n+1));(10*(x%10^(n+1)))\10^(n+1)} \\ Paul D. Hanna, Aug 12 2006

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

A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

Original entry on oeis.org

1, 0, -1, 2, 0, -16, 65, -78, -749, 6232, -22068, -28920, 1004685, -7408740, 22263215, 157632230, -2874256740, 21590948480, -53087332675, -956539294506, 16344490525835, -132605481091060, 294656170409328, 9113173803517344, -167298122286332823

Offset: 0

Mélika Tebni, Jun 21 2021

sign

For all p prime, a(p)/(p-1) == 1 (mod p). - Mélika Tebni, Mar 21 2022

			exp(-1+(x+1)*exp(-x)) = 1 - x^2/2! + 2*x^3/3! - 16*x^5/5! + 65*x^6/6! - 78*x^7/7! - 749*x^8/8! + 6232*x^9/9! + ...

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

Cf. A003725, A014182, A327006.
Cf. A292935 (without 1+x: EGF e^(e^(-x)-1)), A000110 (absolute values: Bell numbers, EGF e^(e^x-1))
Cf. A003725, A106828, A347210, A347571.

a := series(exp(-1+(x+1)*exp(-x)), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
a := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end: seq(a(n), n=0..24);
# third program:
A345652 := n -> add((-1)^(n-k)*combinat[bell](k)*A106828(n, k), k=0..iquo(n, 2)):
seq(A345652(n), n=0..24); # Mélika Tebni, Sep 21 2021

nmax = 24; CoefficientList[Series[Exp[-1+(x+1)*Exp[-x]], {x, 0, nmax}], x] Range[0, nmax]!

seq(n) = {Vec(serlaplace(exp(-1+(x+1)*exp(-x + O(x*x^n)))))} \\ Andrew Howroyd, Jun 21 2021

a(n) = if(n==0, 1, sum(k=2, n, (-1)^(k-1)*(k-1)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Mar 15 2022

The e.g.f. y(x) satisfies y' = -x*y*exp(-x).
a(n) = Sum_{k=0..n-2} (n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k) for n > 0.
Conjecture: a(n) = 0 for only n = 1 and n = 4.
Conjecture: For all p prime, a(p)^2 == 1 (mod p).
Stronger conjecture: For n > 1, a(n) == -1 (mod n) iff n is a prime or 6. - M. F. Hasler, Jun 23 2021
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*Bell(k)*A106828(n, k). - Mélika Tebni, Sep 21 2021
a(n) = Sum_{k=0..n} (-1)^k*A003725(n-k)*Bell(k)*binomial(n, k). - Mélika Tebni, Mar 21 2022

A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 10, 0, 0, 0, 0, 1, 25, 15, 0, 0, 0, 0, 1, 56, 105, 0, 0, 0, 0, 0, 1, 119, 490, 105, 0, 0, 0, 0, 0, 1, 246, 1918, 1260, 0, 0, 0, 0, 0, 0, 1, 501, 6825, 9450, 945, 0, 0, 0, 0, 0, 0, 1, 1012, 22935, 56980, 17325, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Nov 25 2022

{{n, k}} are the number of k-quotient sets of an n-set having at least two elements in each equivalence class. This is the definition and notation (doubling the stacked delimiters of the Stirling set numbers) as given by Fekete (see link).
The formal definition expresses the second order Stirling set numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of second kind' used elsewhere should be dropped in favor of the more systematic one used here.
Also the Bell transform of sign(n) for n >= 0. For the definition of the Bell transform see A264428.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 0;
[2] 0, 1,   0;
[3] 0, 1,   0,    0;
[4] 0, 1,   3,    0,    0;
[5] 0, 1,  10,    0,    0,  0;
[6] 0, 1,  25,   15,    0,  0,  0;
[7] 0, 1,  56,  105,    0,  0,  0,  0;
[8] 0, 1, 119,  490,  105,  0,  0,  0,  0;
[9] 0, 1, 246, 1918, 1260,  0,  0,  0,  0,  0;

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.

Antal E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

A008299 is an irregular subtriangle with more information.
A358622 (second order Stirling cycle numbers).
Cf. A000296 (row sums), alternating row sums (apart from sign): A000587, A293037, and A014182.
Cf. A048993, A201637, A264428, A269939, A340264.

T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j)*(-1)^(k - j),
j = 0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Using the e.g.f.:
egf := exp(z*(exp(t) - t - 1)): ser := series(egf, t, 12):
seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k = 0..n)), n = 0..9);
# Using second order Eulerian numbers:
A358623 := proc(n, k) if n = 0 then return 1 fi;
add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, n - k - j - 1), j = 0..n-k-1)
end: seq(seq(A358623(n, k), k = 0..n), n = 0..11);

# recursion over rows
from functools import cache
@cache
def StirlingSetOrd2(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 0]
    rov: list[int] = StirlingSetOrd2(n - 2)
    row: list[int] = StirlingSetOrd2(n - 1) + [0]
    for k in range(1, n // 2 + 1):
        row[k] = (n - 1) * rov[k - 1] + k * row[k]
    return row
for n in range(9): print(StirlingSetOrd2(n))
# Alternative, using function BellMatrix from A264428.
def f(k: int) -> int:
    return 1 if k > 0 else 0
print(BellMatrix(f, 9))

T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(j, k - j)*<>, where <> denote the second order Eulerian numbers (extending Knuth's notation).
T(n, k) = n!*[z^k][t^n] exp(z*(exp(t) - t - 1)).
T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(n, k - j)*{n - k + j, j}, where {n, k} denotes the Stirling set numbers.
T(n, k) = (n - 1) * T(n-2, k-1) + k * T(n-1, k) with suitable boundary conditions.
T(n + k, k) = A269939(n, k), which might be called the Ward set numbers.

A174530 Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.

Original entry on oeis.org

-1, 0, 3, 4, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79

Offset: 0

Paul Curtz, Mar 21 2010

Filling the top row of a table with T(0,k) = 1/k!, k>=0, the Akiyama-Tanigawa algorithm constructs the following table T(n,k) of fractions, n>=0, k>=0:
1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880,...
0, 1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880, ...
-1, 0, 3/2, 4/3, 5/8, 1/5, 7/144, 1/105, 1/640, 1/4536, 11/403200, ...
-1, -3, 1/2, 17/6, 17/8, 109/120, 197/720, 107/1680, 487/40320, ..
2, -7, -7, 17/6, 73/12, 457/120, 529/360, 2081/5040, 263/2880,...
9, 0, -59/2, -13, 91/8, 421/30, 355/48, 2161/840, 3871/5760, 709/5040, ..
9, 59, -99/2, -195/2, -319/24, 1593/40, 2701/80, 76631/5040, 4285/896,...
The numerators of T(2,k) are the current sequence.
The denominators are 1, 1, 2, 3, 8, 5, 144, 105, 640, 4536, 403200, 332640, 43545600, 37065600,...
T(0,k) = T(1,k+1), shifted.
The left column is T(n,0) = (-1)^(n+1)*A014182(n).
The column T(n,1) appears to be (-1)^n*A074051(n). - R. J. Mathar, Jan 16 2011
a(n) = numerator(A005563(n-1)/(n-1)!), for n>0. - Fred Daniel Kline, Mar 20 2016

D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A089026, A090585, A080305.

nn = 78; Numerator[Simplify[CoefficientList[Series[-Zeta[x] + (Derivative[1][Zeta][x] + x*Derivative[2][Zeta][x])*x, {x, 0, nn}], x]/Table[Derivative[n][Zeta][0], {n, 0, nn}]]] (* Mats Granvik, Nov 11 2013 *)

A379781 a(1)=1, a(2)=2; thereafter, a(n) is the final value at the bottom of the difference triangle of the sequence thus far.

Original entry on oeis.org

1, 2, 1, -2, 0, 7, -14, -12, 155, -408, -364, 7693, -30940, 10712, 637701, -4224222, 9980180, 61922567, -810337234, 4100137008, -958593005, -174952472228, 1662063951016, -6944673371867, -22887336602200, 655644589917172, -5694691183524699, 19946666531550638, 176993602416669640

Offset: 1

Neal Gersh Tolunsky, Jan 02 2025

sign

The difference triangle refers to the triangular array of iterated differences.

The first term in each row of the difference triangle is the inverse binomial transform of the sequence, so the definition means the inverse binomial transform deletes term a(2) = 2.

			To find a(6), we look at the first difference triangle of the first 5 terms:
  1,   2,   1,  -2,   0
  1,  -1,  -3,   2
 -2,  -2,   5
  0,   7
  7
7 is the final value, so a(6)=7.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..595

Cf. A014182, A010739.

a[1] = 1; a[2] = 2; a[n_] := a[n] = Sum[(-1)^(n-k+1) * Binomial[n-2, k-1] * a[k], {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Amiram Eldar, Jan 04 2025 *)

cp's OEIS Frontend

A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + n*j*x).

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A025016 Final digits of !n = Sum_{i=0..n} i! (A003422) for very large n, read from right.

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A153732 Binomial transform of A109747.

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A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

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A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

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A174530 Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.

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A379781 a(1)=1, a(2)=2; thereafter, a(n) is the final value at the bottom of the difference triangle of the sequence thus far.

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A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).