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%I M5155 N2239 #59 Oct 21 2022 22:07:55
%S 24,274,1624,6769,22449,63273,157773,357423,749463,1474473,2749747,
%T 4899622,8394022,13896582,22323822,34916946,53327946,79721796,
%U 116896626,168423871,238810495,333685495,460012995,626334345,843041745,1122686019,1480321269,1933889244
%N Stirling numbers of first kind s(n+4, n).
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 259.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 48.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000915/b000915.txt">Table of n, a(n) for n = 1..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = binomial(n+4, 5)*(15*n^3 + 150*n^2 + 485*n + 502)/48. - _André F. Labossière_, Sep 30 2004
%F Stirling1(n+1, n-3) = Sum_{L=1..n} (Sum_{k=L+1..n} (Sum_{j=k+1..n} (Sum_{i=j+1..n} i*j*k*L))), cf. A001298. - _Vladeta Jovovic_, Jan 31 2005
%F E.g.f. with offset 4: exp(x)*(Sum_{m=0..4} A112486(4,m)*(x^(4+m))/(4+m)!).
%F a(n) = (f(n+3, 4)/8!)*Sum_{m=0..min(4, n-1)} A112486(4,m)*f(8, 4-m)*f(n-1, m), with the falling factorials f(n, m):=n*(n-1)*...*(n-(m-1)).
%F G.f.: x*(24 + 58*x + 22*x^2 + x^3)/(1 - x)^9, see the k=3 row of triangle A112007 for [24, 58, 22, 1].
%F a(n) = A001298(-4-n) for all n in Z. - _Michael Somos_, Sep 04 2017
%p A000915 := proc(n)
%p combinat[stirling1](n+4,n) ;
%p end proc:
%p seq(A000915(n),n=1..10) ; # _R. J. Mathar_, May 19 2016
%t Table[Binomial[n + 4, 5]*(15*n^3 + 150*n^2 + 485*n + 502)/48, {n, 50}] (* _T. D. Noe_, Jun 20 2012 *)
%t a[ n_] := n (n + 1) (n + 2) (n + 3) (n + 4) (15 n^3 + 150 n^2 + 485 n + 502) / 5760; (* _Michael Somos_, Sep 04 2017 *)
%o (PARI) {a(n) = n * (n+1) * (n+2) * (n+3) * (n+4) * (15*n^3+ 150*n^2 + 485*n + 502) / 5760}; /* _Michael Somos_, Sep 04 2017 */
%o (Sage) [stirling_number1(n,n-4) for n in range(5, 30)] # _Zerinvary Lajos_, May 16 2009
%Y Cf. A008275, A094216, A001303 for s(n+3,n), A053567 for s(n+5,n).
%Y Cf. A001298.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
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