A293040 -id:A293040 - 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

A057814 Number of partitions of an n-set into blocks of size > 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 127, 463, 1255, 3004, 6722, 140570, 1039260, 5371627, 23202077, 90048525, 814737785, 7967774337, 62895570839, 417560407223, 2455461090505, 18440499041402, 179627278800426, 1770970802250146

Offset: 0

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

Seiichi Manyama, Table of n, a(n) for n = 0..589 (terms 0..300 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.

Column k=4 of A293024.
Row sums of A059024.
Cf. A000110, A000296, A006505, A057837.
Cf. A293040.

G:={P=Set(Set(Atom,card>=5))}:combstruct[gfsolve](G,labeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..27); # Zerinvary Lajos, Dec 16 2007

max = 27; CoefficientList[ Series[ Exp[ Exp[x] - Normal[ Series[ Exp[x], {x, 0, 4}]]], {x, 0, max}], x]*Range[0, max]!(* Jean-François Alcover, Apr 25 2012, from e.g.f. *)

E.g.f.: 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, Feb 09 2020

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

A293039 E.g.f.: exp(1 + x + x^2/2! + x^3/3! - exp(x)).

Original entry on oeis.org

1, 0, 0, 0, -1, -1, -1, -1, 34, 125, 335, 791, -4027, -41328, -223510, -966174, -1082043, 22493107, 255137121, 1853859145, 8611832136, 6734302429, -364364045001, -4974309134233, -41550393316275, -223452696895652, -173393115915136, 14282249293678744

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

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

Column k=3 of A293051.
Cf. A000587 (k=0), A293037 (k=1), A293038 (k=2), this sequence (k=3), A293040 (k=4).
Cf. A057837.

With[{nn=30},CoefficientList[Series[Exp[1+x+x^2/2!+x^3/3!-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2022 *)

my(x='x+O('x^66)); Vec(serlaplace(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, 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.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A057814 Number of partitions of an n-set into blocks of size > 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A293039 E.g.f.: exp(1 + x + x^2/2! + x^3/3! - exp(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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)).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula