A001702 Generalized Stirling numbers.

1, 24, 154, 580, 1665, 4025, 8624, 16884, 30810, 53130, 87450, 138424, 211939, 315315, 457520, 649400, 903924, 1236444, 1664970, 2210460, 2897125, 3752749, 4809024, 6101900, 7671950, 9564750, 11831274, 14528304, 17718855, 21472615, 25866400, 30984624

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 = 1..1000
• 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.
• 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].
• Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

For n > 1, a(n) = A145324(n+2,4).

Programs

GAP

Concatenation([1],List([2..35],n->(n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48)); # Muniru A Asiru, Sep 29 2018

Magma

[1] cat [n*(n^2-1)*(n+4)*(n^2+7*n+14)/48: n in [2..35]]; // Vincenzo Librandi, Sep 30 2018

Maple

A001702 := proc(n)
    if n = 1 then
        1 ;
    else
        (n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48 ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

Mathematica

Join[{1}, Table[(n-1) n (n+1) (n+4) (n^2 + 7 n + 14)/48, {n, 2, 100}]] (* T. D. Noe, Aug 09 2012 *)
CoefficientList[Series[1 +x*(x-4)*(x^2-2*x+6)/(x-1)^7, {x, 0, 100}], x] (* Stefano Spezia, Sep 30 2018 *)
Join[{1},Table[Coefficient[Product[x + j, {j, 2, k}], x, k - 4], {k, 4, 40}]]  (* or *)  Join[{1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {24, 154, 580, 1665, 4025, 8624, 16884}, 40]] (* Robert A. Russell, Oct 04 2018 *)

PARI

vector(50, n, if(n==1, 1, (1/48)*(n-1)*n* (n+1)* (n+4)*(n^2 +7*n +14))) \\G. C. Greubel, Oct 06 2018

Formula

a(n) = (1/48)*(n-1)*n*(n+1)*(n+4)*(n^2+7n+14), n > 1.

G.f.: x + x^2*(x-4)*(x^2-2*x+6)/(x-1)^7. - Simon Plouffe in his 1992 dissertation

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-1) = -f(n,n-3,2), for n >= 3. - Milan Janjic, Dec 20 2008

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Colin Barker, Jul 08 2020