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A001702 Generalized Stirling numbers.
(Formerly M5148 N2234)
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres reliés aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

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

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

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 *)

PROG

(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

(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

CROSSREFS

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

Sequence in context: A305160 A279459 A092181 * A004308 A008663 A277984

Adjacent sequences:  A001699 A001700 A001701 * A001703 A001704 A001705

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 23 05:13 EDT 2018. Contains 316519 sequences. (Running on oeis4.)