A293039 -id:A293039 - 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

A057837 Number of partitions of a set of n elements where the partitions are of size > 3.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 36, 127, 337, 793, 7525, 48764, 238954, 997790, 6401435, 49107697, 345482807, 2150694855, 14656830110, 116678887407, 978172378669, 7886661080873, 63905475745765, 553437891603452, 5122279358273976, 48331088541366296, 458771027309344261

Offset: 0

Steven C. Fairgrieve (fsteven(AT)math.wvu.edu), Nov 06 2000

Seiichi Manyama, Table of n, a(n) for n = 0..583 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.

Column k=3 of A293024.
Row sums of A059023.
Cf. A000110, A000296, A006505, A057814.
Cf. A293039.

G:={P=Set(Set(Atom,card>=4))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..26); # Zerinvary Lajos, Dec 16 2007

With[{nn=30},CoefficientList[Series[Exp[Exp[x]-1-x-x^2/2-x^3/6],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 28 2012 *)

E.g.f.: exp(exp(x)-1-x-x^2/2-x^3/6).

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

Corrected and extended by Christian G. Bower and James Sellers, Nov 09 2000

A293038 E.g.f.: exp(1 + x + x^2/2! - exp(x)).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, 9, 34, 90, -71, -1645, -9439, -25367, 45902, 1070146, 7122361, 24063637, -54352333, -1501032375, -12319959348, -53177369044, 80189626539, 3910291080509, 40317032441401, 228707685648269, 38882013140648, -16392939262378536

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..592

Column k=2 of A293051.
Cf. A000587 (k=0), A293037 (k=1), this sequence (k=2), A293039 (k=3), A293040 (k=4).
Cf. A006505.

seq(factorial(n) * coeftayl(exp(1+x+x^2/2!-exp(x)), x = 0, n),n = 0..50); # Muniru A Asiru, Oct 05 2017

my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2)))

a(0) = 1; a(n) = -Sum_{k=3..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Nov 20 2020

A293040 E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 125, 461, 1253, 3002, 6720, -111684, -978758, -5246983, -22948029, -89534309, 164027151, 5722510249, 55413784239, 393256686307, 2377996545081, 7807749195198, -46231762188586, -1125536160278906, -12849721017510166

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

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

Column k=4 of A293051.
Cf. A000587 (k=0), A293037 (k=1), A293038 (k=2), A293039 (k=3), this sequence (k=4).
Cf. A057814.

seq(factorial(n)*coeftayl(exp(1+x+x^2/2!+x^3/3!+x^4/4!-exp(x)), x = 0, n),n=0..50); # Muniru A Asiru, Oct 06 2017

my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2+x^3/6+x^4/24)))

a(0) = 1; a(n) = -Sum_{k=5..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Nov 20 2020

A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, 0, -1, 2, -2, 1, 0, 0, 0, -1, 9, -9, 1, 0, 0, 0, -1, -1, 9, -9, 1, 0, 0, 0, 0, -1, 9, -50, 50, 1, 0, 0, 0, 0, -1, -1, 34, -267, 267, 1, 0, 0, 0, 0, 0, -1, -1, 90, -413, 413, 1, 0, 0, 0, 0, 0, -1, -1, 34, -71

Offset: 0

Seiichi Manyama, Sep 29 2017

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1,  0,  0,  0,  0, ...
    0, -1,  0,  0,  0, ...
    1, -1, -1,  0,  0, ...
    1,  2, -1, -1,  0, ...
   -2,  9, -1, -1, -1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000587, A293037, A293038, A293039, A293040.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A229223, A293024.

E.g.f. of column k: Product_{i>k} exp(-x^i/i!).

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

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

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A057837 Number of partitions of a set of n elements where the partitions are of size > 3.

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A293038 E.g.f.: exp(1 + x + x^2/2! - exp(x)).

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A293040 E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).

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A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

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