A001716 - OEIS

%I M4651 N1990 #81 Jan 25 2023 11:50:25

%S 1,9,74,638,5944,60216,662640,7893840,101378880,1397759040,

%T 20606463360,323626665600,5395972377600,95218662067200,

%U 1773217155225600,34758188233574400,715437948072960000,15429680577561600000,347968129734973440000,8190600438533990400000

%N Generalized Stirling numbers.

%C The asymptotic expansion of the higher order exponential integral E(x,m=2,n=4) ~ exp(-x)/x^2*(1 - 9/x + 74/x^2 - 638/x^3 + 5944/x^4 - 60216/x^5 + 662640/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - _Johannes W. Meijer_, Oct 20 2009

%C From _Petros Hadjicostas_, Jun 23 2020: (Start)

%C For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and Mitrinovic and Mitrinovic (1962) using slightly different notation.

%C These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0.

%C As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_0^0(a,b) = 1, R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m.

%C With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m) for n, m >= 0.

%C We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.

%C For the current sequence, a(n) = R_{n+1}^1(a=-4, b=-1) for n >= 0. (End)

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001716/b001716.txt">Table of n, a(n) for n = 0..100</a>

%H D. S. Mitrinovic, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k762d/f996.image.r=1961%20mitrinovic">Sur une classe de nombres reliés aux nombres de Stirling</a>, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are introduced.]

%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43667130">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].

%H D. S. Mitrinovic and M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.

%H J. Riordan, <a href="/A000254/a000254.pdf">Letter of 04/11/74</a>.

%F a(n) = Sum_{k=0..n} (-1)^(n+k) * (k+1) * 4^k * stirling1(n+1, k+1). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

%F a(n-1) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-4,k)/(n-k) for n >= 1. [_Milan Janjic_, Dec 14 2008] [Edited by _Petros Hadjicostas_, Jun 23 2020]

%F a(n)= n! * [3]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n (with offset 1). [_Gary Detlefs_, Jan 04 2011]

%F a(n) = (n+1)! * Sum_{k=0..n} (-1)^k*binomial(-4,k)/(n+1-k). [_Gary Detlefs_, Jul 16 2011]

%F a(n) = (n+4)! * Sum_{k=1..n+1} 1/(k+3)/6. [_Gary Detlefs_, Sep 14 2011]

%F E.g.f. (for offset 1): 1/(1-x)^4 * log(1/(1-x)). - _Vaclav Kotesovec_, Jan 19 2014

%F E.g.f.: (1 + 4*log(1/(1 - x)))/(1 - x)^5. - _Ilya Gutkovskiy_, Jan 23 2017

%F From _Petros Hadjicostas_, Jun 23 2020: (Start)

%F a(n) = [x] Product_{r=0..n} (x + 4 + r) = (Product_{r=0..n} (4 + r)) * Sum_{i=0..n} 1/(4 + i).

%F Since a(n) = R_{n+1}^1(a=-4, b=-1) and R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b), we conclude that:

%F (i) a(n) = (n+3)!/6 + (n+4)*a(n-1) for n >= 1;

%F (ii) a(n) = (2*n+7)*a(n-1) - (n+3)^2*a(n-2) for n >= 2. (End)

%t f[k_] := k + 3; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}] (* _Clark Kimberling_, Dec 29 2011 *)

%t Rest[CoefficientList[Series[(1-x)^(-4)*Log[1/(1-x)],{x,0,20}],x]*Range[0,20]!] (* _Vaclav Kotesovec_, Jan 19 2014 *)

%o (PARI) R(n, m, a, b) =  sum(k=0, n-m, (-1)^k*a^k*b^(n-m-k)*binomial(m+k, k)*stirling(n, m+k, 1));

%o aa(n) = R(n+1, 1, -4, -1);

%o for(n=0, 19, print1(aa(n), ", ")) \\ _Petros Hadjicostas_, Jun 23 2020

%Y Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004