A245109 -id:A245109 - cp's OEIS frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

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

Offset: 0

José María Grau Ribas, Jun 17 2013

			-0.4063757399599599076769581241248397582109975751811406350004954883....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.

RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)

solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017

Equals -2*A106533.

Equals LambertW(-2*exp(-2)).

A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 15, 37, 31, 10, 1, 52, 151, 160, 75, 15, 1, 203, 674, 856, 520, 155, 21, 1, 877, 3263, 4802, 3556, 1400, 287, 28, 1, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 1, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1

Offset: 0

N. J. A. Sloane

Triangle a(n,k) read by rows; given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. - Paul Barry, Jan 12 2009
Equal to A048993*A007318. - Philippe Deléham, Oct 31 2011
This lower unitriangular array is the L factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20U%20factor%20is%20A059098%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The U factor is A059098 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   5, 10,  6,  1;
  15, 37, 31, 10,  1;
  ...
From _Paul Barry_, Jan 12 2009: (Start)
Production array begins
  1, 1;
  1, 2, 1;
  0, 2, 3, 1;
  0, 0, 3, 4, 1;
  0, 0, 0, 4, 5, 1;
  ... (End)

Alois P. Heinz, Rows n = 0..140, flattened
M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 11.
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 15 Feb. 2013, pp. 1667-1677.
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.
J. Riordan, Letter, Oct 31 1977. The array is on the second page.

First column gives A000110, second column = A005493.
Third column = A003128, row sums = A001861, A059340.
See A244489 for another version of this triangle.
Cf. A059098, A245109, A046716.

a:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, a(n-1, k-1) +(k+1)*(a(n-1, k) +a(n-1, k+1))))
    end:
seq(seq(a(n, k), k=0..n), n=0..15);  # Alois P. Heinz, Nov 30 2012

a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]]
(* Jean-François Alcover, Aug 29 2011, after Vladeta Jovovic *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n),k))

def A049020_triangle(dim):
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k]+(k+1)*M[n-1, k+1]
    return M
A049020_triangle(9) # Peter Luschny, Sep 19 2012

a(n,k) = a(n-1, k-1) + (k+1)*a(n-1, k) + (k+1)*a(n-1, k+1), n >= 1.
a(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(i,k). - Vladeta Jovovic, Jan 27 2001
E.g.f. for the k-th column is (1/k!)*(exp(x)-1)^k*exp(exp(x)-1). - Vladeta Jovovic, Jan 27 2001
G.f.: 1/(1-x-xy-x^2(1+y)/(1-2x-xy-2x^2(1+y)/(1-3x-xy-3x^2(1+y)/(1-4x-xy-4x^2(1+y)/(1-... (continued fraction). - Paul Barry, Apr 29 2009
E.g.f.: exp((y+1)*(exp(x)-1)). - Geoffrey Critzer, Nov 30 2012
Note that A244489 (which is essentially the same triangle) has the formula T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1. - N. J. A. Sloane, May 17 2016
a(2n,n) = A245109(n). - Alois P. Heinz, Aug 23 2017
Empirical: a(n,k) = p(1^n)[st(1^k)] (see A002872 for notation). - Andrey Zabolotskiy, Oct 17 2017
a(n,k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j)/k!. - Peter Luschny, Dec 06 2023

More terms from James Sellers.

Better definition from Geoffrey Critzer, Nov 30 2012.

A245110 G.f.: Sum_{n>=0} ( exp(-1/(1-nx)) / (1-nx)^n ) / n!.

Original entry on oeis.org

1, 1, 4, 23, 161, 1302, 11810, 117889, 1277890, 14894043, 185226966, 2442933979, 33998594943, 497207012018, 7613797641286, 121711037138949, 2025687745708717, 35020194893837462, 627586143525936866, 11636932722633705392, 222893347544826491780, 4403534468187986689781

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Compare the g.f. to: Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 161*x^4 + 1302*x^5 + 11810*x^6 +...
where
A(x) = exp(-1) + exp(-1/(1-x))/(1-x) + (exp(-1/(1-2*x))/(1-2*x)^2)/2!
+ (exp(-1/(1-3*x))/(1-3*x)^3)/3! + (exp(-1/(1-4*x))/(1-4*x)^4)/4!
+ (exp(-1/(1-5*x))/(1-5*x)^5)/5! + (exp(-1/(1-6*x))/(1-6*x)^6)/6!
+ (exp(-1/(1-7*x))/(1-7*x)^7)/7! + (exp(-1/(1-8*x))/(1-8*x)^8)/8! +...
simplifies to a power series in x with integer coefficients.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A245109, A245111.

/* From definition (requires setting suitable precision) */ \p100
{a(n)=local(A=1+x, X=x+x*O(x^n)); A=suminf(k=0, exp(-1/(1-k*X))/(1-k*X)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

/* From a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) */
{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0,1,sum(k=1,n,Stirling2(n, k) * binomial(n+k-1, k-1)))}
for(n=0, 30, print1(a(n),", "))

/* As row sums of triangle A245111: */
{A245111(n,k)=local(A=1+x*y); A=sum(k=0, n, 1/(1-k*x+x*O(x^n))^k*y^k/k!*exp(-y/(1-k*x+x*O(x^n))+y*O(y^n))); polcoeff(polcoeff(A, n,x),k,y)}
{a(n) = sum(k=0,n, A245111(n,k))}
/* Print Initial Rows of Triangle A245111: */
{for(n=0, 10, for(k=0,n, print1(A245111(n,k),", "));print(""))}
/* Row Sums yield A245110: */
for(n=0, 30, print1(a(n),", "))

a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) for n>0 with a(0)=1.

Row sums of Triangle A245111.

A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

Original entry on oeis.org

1, 2, 12, 130, 2216, 52078, 1558219, 56524414, 2406802476, 117575627562, 6478447651345, 397345158550386, 26842747368209994, 1980156804133210116, 158365138356099680582, 13647670818304698139989, 1260732993182758276252088, 124273946254095006307105363

Offset: 0

Alois P. Heinz, May 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A256130, A187655, A187657, A217899, A217900, A245109, A319732.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(2*n, n):
seq(a(n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := T[2n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

a(n) = A256130(2n,n).
a(n) ~ 2^(2*n-1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 31 2015
a(n) ~ Stirling2(2*n, n) = A007820(n). - Vaclav Kotesovec, Jun 01 2015

cp's OEIS Frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

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A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

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A245110 G.f.: Sum_{n>=0} ( exp(-1/(1-n*x)) / (1-n*x)^n ) / n!.

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A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

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Author

Keywords

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Crossrefs

Programs

Maple

Mathematica

Formula

A245110 G.f.: Sum_{n>=0} ( exp(-1/(1-nx)) / (1-nx)^n ) / n!.