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 A040027 The Gould numbers. 25
 1, 1, 3, 9, 31, 121, 523, 2469, 12611, 69161, 404663, 2512769, 16485691, 113842301, 824723643, 6249805129, 49416246911, 406754704841, 3478340425563, 30845565317189, 283187362333331, 2687568043654521, 26329932233283223 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..500 W. Asakly, A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, S. 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. Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017. Branko Dragovich, Andrei Yu. Khrennikov, Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079 [math.NT], 2015. B. Dragovich, 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] S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003). S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002. Don Knuth, Email to N. J. A. Sloane, Jan 29 2018 FORMULA 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)) * int[0..x, t*exp(-exp(t)+t) dt] (fits the recurrence up to n=215). - Ralf Stephan, Apr 25 2004 Recurrence : a(1)=1, a(2)=1, for n>2, a(n)=n-1+sum(j=2, n-1, binomial(n-1, j)*a(j)) [gives a(n+1)]. - 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) EXAMPLE 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 MAPLE 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 MATHEMATICA a = a = 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 *) PROG (PARI) {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 */ (Haskell) a040027 n = head \$ a046936_row (n + 1)  -- Reinhard Zumkeller, Jan 01 2014 (Python) # 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 CROSSREFS Left-hand border of triangle A046936. Cf. also A011971, 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). Sequence in context: A066571 A087648 A086616 * A182968 A071603 A090595 Adjacent sequences:  A040024 A040025 A040026 * A040028 A040029 A040030 KEYWORD easy,nonn,nice AUTHOR EXTENSIONS 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 STATUS approved

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Last modified April 18 01:28 EDT 2021. Contains 343072 sequences. (Running on oeis4.)