A108458 -id:A108458 - cp's OEIS frontend

A040027 The Gould numbers.

1, 1, 3, 9, 31, 121, 523, 2469, 12611, 69161, 404663, 2512769, 16485691, 113842301, 824723643, 6249805129, 49416246911, 406754704841, 3478340425563, 30845565317189, 283187362333331, 2687568043654521, 26329932233283223, 265946395403810289, 2766211109503317451

Offset: 0

Henry Gould

Number of permutations beginning with 21 and avoiding 1-23. - Ralf Stephan, Apr 25 2004
Originally defined as main diagonal of an array of binomial recurrence coefficients (see Gould and Quaintance). Also second-from-right diagonal of triangle A121207.
Starting (1, 3, 9, 31, 121, ...) = row sums of triangle A153868. - Gary W. Adamson, Jan 03 2009
Equals eigensequence of triangle A074909 (reflected). - Gary W. Adamson, Apr 10 2009
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m=>-1, is related to the sequence given above. For m=-1 this series dates back to Euler. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A073003 Gompertz's constant and A000110 the Bell numbers, see A163940; A040027(m = -1) = 0. - Johannes W. Meijer, Oct 16 2009
Compare the o.g.f. to the o.g.f. B(x) of the Bell numbers, where B(x) = 1 + x*B(x/(1-x))/(1-x). - Paul D. Hanna, Mar 23 2012
a(n) is the number of set partitions of {1,2,...,n+1} in which the last block is a singleton: the blocks are arranged in order of their least element. An example is given below. - Peter Bala, Dec 17 2014

			a(3) = 9: Arranging the blocks of the 15 set partitions of {1,2,3,4} in order of their least element we find 9 set partitions for which the last block is a singleton, namely, 123|4, 124|3, 134|2, 1|24|3, 1|23|4, 12|3|4, 13|2|4, 14|2|3, and 1|2|3|4. - _Peter Bala_, Dec 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Walaa Asakly, Aubrey Blecher, Charlotte Brennan, Arnold Knowfmacher, Toufik Mansour, and Stephan Wagner, Set partition asymptotics and a conjecture of Gould and Quaintance, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2, 15 August 2014, Pages 672-682.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See pp. 8-9,17.
Robert Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv:2210.13520 [cs.SC], 2022.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 69.
Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079 [math.NT], 2015.
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, October 2014, Volume 6, Issue 4, pp 275-283.
H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials, Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968 [The letter gives the g.f. for this sequence as e^{e^x} Integral_{0..x} e^{e^t-1} dt but the correct g.f. is e^{e^x-1} Integral_0^x e^{1-e^t} dt. - _Don Knuth_, Feb 01 2018]
Sergey Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
Sergey Kitaev and Toufik Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Left-hand border of triangle A046936. Cf. also A011971, A014619, A298804.
Cf. A153868. - Gary W. Adamson, Jan 03 2009
Cf. A074909. - Gary W. Adamson, Apr 10 2009
Row sums of A163940. - Johannes W. Meijer, Oct 16 2009
Cf. A108458 (row sums), A124496 (column 1).

a040027 n = head $ a046936_row (n + 1)  -- Reinhard Zumkeller, Jan 01 2014

A040027 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add(binomial(n,k-1)*procname(n-k),k=1..n) ;
    end if;
end proc: # Johannes W. Meijer, Oct 16 2009

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n, k + 1]*a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]  (* Jean-François Alcover, Jul 02 2013 *)
Rest[CoefficientList[Assuming[Element[x, Reals], Series[E^E^x*(ExpIntegralEi[-E^x] - ExpIntegralEi[-1]), {x, 0, 20}]], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 28 2014 *)

{a(n)=local(A=1+x);for(i=1,n,A=1+x*subst(A,x,x/(1-x+x*O(x^n)))/(1-x)^2);polcoeff(A,n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A040027_list = lambda size: Gould_diag(2, size)
print(A040027_list(24)) # Peter Luschny, Apr 24 2016

a(n) = b(n-2), n>1, b(n) = Sum_{k = 1..n} binomial(n, k-1)*b(n-k), b(0) = 1. - Vladeta Jovovic, Apr 28 2001
E.g.f. satisfies A'(x) = exp(x)*A(x)+1. - N. J. A. Sloane
With offset 0, e.g.f.: x + exp(exp(x)) * Integral_{t=0..x} t*exp(-exp(t)+t) dt (fits the recurrence up to n=215). - Ralf Stephan, Apr 25 2004
Recurrence: a(0)=1, a(1)=1, for n > 1, a(n) = n + Sum_{j=1..n-1} binomial(n, j+1)*a(j). - Jon Perry, Apr 26 2005
O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^2. - Paul D. Hanna, Mar 23 2012
From Peter Bala, Dec 17 2014: (Start)
Starting from A(x) = 1 + O(x) (big Oh notation) we can get a series expansion for the o.g.f. by repeatedly applying the above functional equation of Hanna: A(x) = 1 + O(x) = 1 + x/(1-x)^2 + O(x^2) = 1 + x/(1-x)^2 + x^2/((1-x)*(1-2*x)^2) + O(x^3) = ... = 1 + x/(1-x)^2 + x^2/((1-x)*(1-2*x)^2) + x^3/((1-x)*(1-2*x)*(1-3*x)^2) + x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)^2) + ....
a(n) = Sum_{k = 0..n} ( Sum_{j = k..n} Stirling2(j,k)*k^(n-j) ).
Row sums of A108458. First column of A124496. (End)
Conjecture: a(n) = Sum_{k = 0..n} A058006(k)*A048993(n+1, k+1) - Velin Yanev, Aug 31 2021

Entry revised by N. J. A. Sloane, Dec 11 2006
Gould reference updated by Johannes W. Meijer, Aug 02 2009
Don Knuth, Jan 29 2018, suggested that this sequence should be named after H. W. Gould. - N. J. A. Sloane, Jan 30 2018

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

Original entry on oeis.org

1, 1, 5, 52, 855, 19921, 614866, 24040451, 1152972925, 66200911138, 4465023867757, 348383154017581, 31052765897026352, 3128792250765898965, 353179564583216567917, 44320731930172534543092, 6141797839043095806714667, 934330605640859569909566925

Offset: 0

Christian G. Bower, Jun 03 2005

nonn

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one white object.

a(n) is also the number of elements of the partition monoid P_n with domain {1,...,n}. Elements of P_n are set partitions of {1,1',...,n,n'}, and the domain of such a partition is the set of all points in {1,...,n} that belong to a block containing a dashed element. - James East, Apr 10 2018

Seiichi Manyama, Table of n, a(n) for n = 0..200

Main diagonal of A108458. Cf. A108461.
Cf. A048993 (Stirling2), A068424 (falling factorial).
Cf. A326600, A326270, A326271, A326288.
Bisection of A124421 (even part).

b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 02 2023

a[n_] := If[n == 0, 1, Sum[k^n*StirlingS2[n, k], {k, 0, n}]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2024 *)

{a(n)=polcoeff(sum(m=0, n, m^m*x^m/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 17 2013

{a(n)=n!*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!), n)} \\ Paul D. Hanna, Sep 17 2013

a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - Vladeta Jovovic, Aug 31 2006
E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - Vladeta Jovovic, Jul 12 2007
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - Paul D. Hanna, Jun 28 2019
O.g.f.: Sum_{n>=0} n^n * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Sep 17 2013
a(n) = Sum_{k=0..n} Stirling2(n,k) * Sum_{l=k..n} Stirling2(n,l)*T(l,k). Here T(l,k) are the falling factorials. - James East, Apr 10 2018

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A040027 The Gould numbers.

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A108459 Number of labeled partitions of (n,n) into pairs (i,j).

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