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A001718 Generalized Stirling numbers.

%I A001718 M5127 N2222 #43 Jun 27 2020 16:23:00
%S A001718 1,22,355,5265,77224,1155420,17893196,288843260,4876196776,
%T A001718 86194186584,1595481972864,30908820004608,626110382381184,
%U A001718 13246845128678016,292374329134060800,6723367631258860800,160883166944083161600,4001062259532015244800
%N A001718 Generalized Stirling numbers.
%C A001718 The asymptotic expansion of the higher order exponential integral E(x,m=4,n=4) ~ exp(-x)/x^4*(1 - 22/x + 355/x^2 - 5265/x^3 + 77224/x^4 - 1155420/x^5 + ...) leads to the sequence given above. See A163931 and A163934 for more information. - _Johannes W. Meijer_, Oct 20 2009
%C A001718 From _Petros Hadjicostas_, Jun 26 2020: (Start)
%C A001718 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 A001718 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 A001718 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 A001718 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 A001718 For the current sequence, a(n) = R_{n+3}^3(a=-4, b=-1) for n >= 0. (End)
%D A001718 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001718 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001718 T. D. Noe, <a href="/A001718/b001718.txt">Table of n, a(n) for n = 0..100</a>
%H A001718 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 A001718 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 A001718 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.
%F A001718 a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(k+3, 3) * 4^k * Stirling1(n+3, k+3). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
%F A001718 E.g.f.: (1 - 15*log(1 - x) + 37*log(1 - x)^2 - 20*log(1 - x)^3)/(1 - x)^7. - _Vladeta Jovovic_, Mar 01 2004
%F A001718 If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k) * Stirling1(n-k,i) * Product_{j=0..k-1} (-a-j), then a(n-3) = |f(n,3,4)| for n >= 3. - _Milan Janjic_, Dec 21 2008
%F A001718 From _Petros Hadjicostas_, Jun 26 2020: (Start)
%F A001718 a(n) = [x^3] Product_{r=0..n+2} (x + 4 + r) = (Product_{r=0..n+2} (4 + r)) * Sum_{0 <= i < j < k <= n+2} 1/((4 + i)*(4 + j)*(4 + k)).
%F A001718 E.g.f.: Sum_{n >= 0} a(n)/(n+3)!*x^(n+3) = -(log(1 - x))^3/(6*(1 - x)^4).
%F A001718 Since a(n) = R_{n+3}^3(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 A001718 (i) a(n) = A001717(n) + (n+6)*a(n-1) for n >= 1;
%F A001718 (ii) a(n) = A001716(n) + (2*n+11)*a(n-1) - (n+5)^2*a(n-2) for n >= 2.
%F A001718 (iii) a(n) = (n+3)!/6 + 3*(n+5)*a(n-1) - (3*n^2+27*n+61)*a(n-2) + (n+4)^3*a(n-3) for n >= 3.
%F A001718 (iv) a(n) = 2*(2*n+9)*a(n-1) - (6*n^2+48*n+97)*a(n-2) + (2*n+7)*(2*n^2+14*n+25)*a(n-3) - (n+3)^4*a(n-4) = 0 for n >= 4. (End)
%t A001718 nn = 20; t = Range[0, nn]! CoefficientList[Series[(1 - 15*Log[1 - x] + 37*Log[1 - x]^2 - 20*Log[1 - x]^3)/(1 - x)^7, {x, 0, nn}], x] (* _T. D. Noe_, Aug 09 2012 *)
%o A001718 (PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+3, 3)*4^k*stirling(n+3, k+3, 1)); \\ _Michel Marcus_, Jan 20 2016
%Y A001718 Cf. A001716, A001717, A163931, A163934.
%K A001718 nonn
%O A001718 0,2
%A A001718 _N. J. A. Sloane_
%E A001718 More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004