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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
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
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
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)