A193684 -id:A193684 - 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

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

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).

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

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

A367819 Expansion of e.g.f. exp(1 - 4*x - exp(x)).

Original entry on oeis.org

1, -5, 24, -111, 497, -2166, 9239, -38765, 160658, -659773, 2691205, -10922544, 44166173, -178098121, 716703848, -2879774019, 11558005677, -46348854134, 185746261419, -744036460097, 2979305960426, -11926715433881, 47735079979633, -191026723545976, 764362047956073, -3058170811731677

Offset: 0

Ilya Gutkovskiy, Dec 01 2023

sign

Cf. A000587, A045379, A109747, A153732, A193684, A346739, A367818.

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

my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 4*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

G.f. A(x) satisfies: A(x) = 1 - 4*x*A(x) - x * A(x/(1 - x)) / (1 - x).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-4)^n / k!.
a(0) = 1; a(n) = -4*a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A000587(k).

cp's OEIS Frontend

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

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

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

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

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

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A367819 Expansion of e.g.f. exp(1 - 4*x - exp(x)).

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