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%I #23 Nov 14 2019 12:33:27
%S 1,1,2,7,8,3,6,38,93,111,65,15,24,226,874,1821,2224,1600,630,105,120,
%T 1524,8200,24860,47185,58465,47474,24430,7245,945,720,11628,81080,
%U 326712,852690,1522375,1905168,1676325,1018682,407925,97020,10395,5040
%N Triangle read by rows giving the coefficients of formulas generating each variety of S1(n,k) (unsigned Stirling numbers of first kind). The p-th row (p>=1) contains T(i,p) for i=1 to 2*p, where T(i,p) satisfies Sum_{i=1..2*p} T(i,p) * C(n,i).
%C The formulas S1(n+p,n) obtained are those of S1(n+2,n) { A000914 }, S1(n+3,n) { A001303 }, S1(n+4,n) { A000915 }, S1(n+5,n) { A053567 } and so on.
%D Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964, 9th Printing (1970), pp. 833-834.
%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 Francis L. Miksa (1901-1975), <a href="https://www.jstor.org/stable/2002617">Stirling numbers of the first kind</a>, "27 leaves reproduced from typewritten manuscript on deposit in the UMT File", Mathematical Tables and Other Aids to Computation, vol. 10, no. 53, January 1956, pp. 37-38 (Reviews and Descriptions of Tables and Books, 7[I]).
%H Dragoslav S. Mitrinovic (1908-1995), <a href="http://pefmath2.etf.bg.ac.rs/files/23/23.pdf">Sur les nombres de Stirling de première espèce et les polynômes de Stirling</a>, AMS 11B73_05A19, Publications de la Faculté d'Electrotechnique de l'Université de Belgrade, Série Mathématiques et Physique (ISSN 0522-8441), no. 23, 1959 (5.V.1959), pp. 1-20.
%H John J. O'Connor and Edmund F. Robertson, <a href="http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Stirling.html">James Stirling (1692-1770)</a>, (September 1998).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling numbers of the first kind</a>.
%H Stephen Wolfram, Wolfram Research, Mathematica 5.2, <a href="http://library.wolfram.com/webMathematica/Education/LongDivide.jsp">webMathematica 2</a>.
%F a(1,k) = k!
%F ...
%F a(2*k-5,k) = a(2*k,k) * (175000*k^8 -2117500*k^7 +10856650*k^6 -30743377*k^5 +52511770*k^4 -55386931*k^3 +35321832*k^2 -12560580*k+1944000) / (1632960*k^3 -7348320*k^2 +9389520*k -3061800).
%F a(2*k-4,k) = a(2*k,k) * (2500*k^6 -17400*k^5 +48511*k^4 -69378*k^3 +53929*k^2 -21906*k +3744) / (7776*k^2-15552*k+5832).
%F a(2*k-3,k) = a(2*k,k) * (1250*k^4-4225*k^3+5023*k^2-2600*k+528) / (1620*k-810).
%F a(2*k-2,k) = a(2*k,k) * (50*k^3-93*k^2+55*k-12) / (36*k-18).
%F a(2*k-1,k) = a(2*k,k) * (5*k-2) / 3.
%F a(2*k,k) = (2*k)! / (k!*2^k).
%e Row 5 contains 120,1524,8200,24860,47185,58465,47474,24430,7245,945, so the formula generating S1(n+5,n) numbers { A053567 } will be the following : 120*n +1524*C(n,2) +8200*C(n,3) +24860*C(n,4) +47185*C(n,5) +58465*C(n,6) +47474*C(n,7) +24430*C(n,8) +7245*C(n,9) +945*C(n,10). And then substituting for the 10th number of such a S1(n+p,n) gives S1(15,10) = 37312275.
%t row[m_] := Module[{eq, t}, eq[n_] := Array[t, 2 m].Table[Binomial[n, k], {k, 1, 2 m}] == Abs[StirlingS1[n + m, n]]; Array[t, 2 m] /. Solve[ Array[ eq, 2 m]] // First];
%t Array[row, 7] // Flatten (* _Jean-François Alcover_, Nov 14 2019 *)
%Y Cf. A000914, A001303, A000915, A053567, A008275, A008276.
%Y Cf. A000012, A000217, A001147, A000142, A094262.
%K easy,nonn,tabl
%O 1,3
%A _André F. Labossière_, May 27 2004, Feb 21 2007
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