A001717 Generalized Stirling numbers. (Formerly M4984 N2143)

1, 15, 179, 2070, 24574, 305956, 4028156, 56231712, 832391136, 13051234944, 216374987520, 3785626465920, 69751622298240, 1350747863435520, 27437426560500480, 583506719443584000, 12969079056388224000, 300749419818102528000, 7265204785551331584000

Offset

0,2

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=4) ~ exp(-x)/x^3*(1 - 15/x + 179/x^2 - 2070/x^3 + 24574/x^4 - 305956/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009

From Petros Hadjicostas, Jun 25 2020: (Start)

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.

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.

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.

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.

For the current sequence, a(n) = R_{n+2}^2(a=-4, b=-1) for n >= 0. (End)

References

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

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

Links

T. D. Noe, Table of n, a(n) for n = 0..100

D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are introduced.]

D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].

D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.

Formula

a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(k+2, 2) * 4^k * Stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

E.g.f.: (1 - 9*log(1 - x) + 10*log(1 - x)^2)/(1 - x)^6. - Vladeta Jovovic, Mar 01 2004

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-2) = |f(n,2,4)| for n>=2. - Milan Janjic, Dec 21 2008

From Petros Hadjicostas, Jun 26 2020: (Start)

a(n) = [x^2] Product_{r=0..n+1} (x + 4 + r) = (Product_{r=0..n+1} (4 + r)) * Sum_{0 <= i < j <= n+1} 1/((4 + i)*(4 + j)).

Since a(n) = R_{n+2}^2(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:

(i) a(n) = A001716(n) + (n+5)*a(n-1) for n >= 1;

(ii) a(n) = (n+3)!/6 + (2*n+9)*a(n-1) - (n+4)^2*a(n-2) for n >= 2.

(iii) a(n) = 3*(n+4)*a(n-1) - (3*n^2+21*n+37)*a(n-2) + (n+3)^3*a(n-3) for n >= 3. (End)

Mathematica

nn = 20; t = Range[0, nn]! CoefficientList[Series[(1 - 9*Log[1 - x] + 10*Log[1 - x]^2)/(1 - x)^6, {x, 0, nn}], x] (* T. D. Noe, Aug 09 2012 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+2, 2)*4^k*stirling(n+2, k+2, 1)); \\ Michel Marcus, Jan 20 2016

Crossrefs

Sequence in context: A012800 A016216 A244604 * A293476 A004992 A055084

Adjacent sequences: A001714 A001715 A001716 * A001718 A001719 A001720

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved