A293038 -id:A293038 - 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

A006505 Number of partitions of an n-set into boxes of size >2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2557, 11353, 60105, 362506, 2169246, 13580815, 91927435, 650078097, 4762023647, 36508923530, 292117087090, 2424048335917, 20847410586719, 185754044235873, 1711253808769653, 16272637428430152

Offset: 0

N. J. A. Sloane

J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..579 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 102
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
J. Riordan, Cached copy of paper [With permission]

Column k=2 of A293024.
Cf. A000110, A000296, A057814, A057837, A059022.
Cf. A293038.

Copy ZL := [ B,{B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; # Zerinvary Lajos, Mar 13 2007
G:={P=Set(Set(Atom,card>=3))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..25); # Zerinvary Lajos, Dec 16 2007
g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1,i)*g(i),i=0..n-3)); end; [seq(subs(x=1,g(n)),n=0..60)]; # N. J. A. Sloane, Jul 20 2011

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* Michael Somos, Jul 20 2011 *)
a[0] = 1; a[n_] := n!*Sum[Sum[k!*(-1)^(m-k)*Binomial[m, k]*Sum[StirlingS2[i+k, k]* Binomial[m-k, n-m-i]*2^(-n+m+i)/(i+k)!, {i, 0, n-m}], {k, 0, m}]/m!, {m, 1, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
Table[Sum[(-1)^j * Binomial[n, j] * BellB[n-j] * 2^((j-1)/2) * HypergeometricU[(1 - j)/2, 3/2, 1/2], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Feb 09 2020 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* Michael Somos, Jul 20 2011 */

E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).
a(n) = Sum_{k=1..[n/3]} A059022(n,k), n>=3. - R. J. Mathar, Nov 08 2008
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]
a(0) = 1; a(n) = Sum_{k=0..n-3} binomial(n-1,k+2) * a(n-k-3). - Seiichi Manyama, Sep 22 2023

More terms from Christian G. Bower, Nov 09 2000

Edited by N. J. A. Sloane, Jul 20 2011

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

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.

A337062 E.g.f.: exp(1 + x^2/2 - exp(x)).

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 21, -51, 113, -498, 1088, -3335, 21407, -14653, 232389, -1275288, -3636526, -44468245, -7468609, 700603965, 12178055777, 67189448344, 175549544778, -2432123216941, -36279392911507, -287078642854853, -945866835928323

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

sign

Cf. A000587, A001147, A097514, A293037, A293038.

nmax = 26; CoefficientList[Series[Exp[1 + x^2/2 - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 26}]
Table[Sum[Binomial[n, 2 k] (2 k - 1)!! BellB[n - 2 k, -1], {k, 0, Floor[n/2]}], {n, 0, 26}]

a(0) = 1; a(n) = -a(n-1) - Sum_{k=3..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * (2*k-1)!! * A000587(n-2*k).

cp's OEIS Frontend

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

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A006505 Number of partitions of an n-set into boxes of size >2.

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A293039 E.g.f.: exp(1 + x + x^2/2! + x^3/3! - 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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A337062 E.g.f.: exp(1 + x^2/2 - exp(x)).

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