Search: id:A094216
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%I A094216
%S A094216 1,1,2,7,8,3,6,38,93,111,65,15,24,226,874,1821,2224,1600,630,105,120,
%T A094216 1524,8200,24860,47185,58465,47474,24430,7245,945,720,11628,81080,
%U A094216 326712,852690,1522375,1905168,1676325,1018682,407925,97020,10395,5040
%N A094216 Triangle read by rows giving the coefficients of formulae 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 A094216 The formulae 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 A094216 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 A094216 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A094216 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
               Sobalian Coefficients</a>.
%H A094216 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               index.html">Miscellaneous</a>.
%H A094216 Francis L. Miksa (1901-1975), <a href="http://links.jstor.org/sici?sici=0891-6837%28195601%2910%3A53%3C35%3AR\
               ADOTA%3E2.0.CO%3B2-X">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 A094216 Dragoslav S. Mitrinovic (1908-1995), <a href="http://pefmath2.etf.bg.ac.yu/
               files/23/23.pdf">Sur les nombres de Stirling de premiere espece et 
               les polynomes de Stirling</a>, AMS 11B73_05A19, Publications de la 
               Faculte d'Electrotechnique de l'Universite de Belgrade, Serie Mathematiques 
               et Physique (ISSN 0522-8441), no. 23, 1959 (5.V.1959), pp. 1-20.
%H A094216 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 A094216 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StirlingNumberoftheFirstKind.html">Stirling numbers of the first 
               kind</a>.
%H A094216 Stephen Wolfram, Wolfram Research, Mathematica 5.2, <a href="http://library.wolfram.com/
               webMathematica/Education/LongDivide.jsp">webMathematica 2</a>.
%F A094216 a(1,k) = k!
%F A094216 ...
%F A094216 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 A094216 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 A094216 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 A094216 a(2*k-2,k) = a(2*k,k) * (50*k^3-93*k^2+55*k-12) / (36*k-18)
%F A094216 a(2*k-1,k) = a(2*k,k) * (5*k-2) / 3
%F A094216 a(2*k,k) = (2*k)! / (k!*2^k).
%e A094216 Row 5 contains 120,1524,8200,24860,47185,58465,47474,24430,7245,945, 
               so the
%e A094216 formula generating S1(n+5,n) numbers { A053567 } will be the following 
               : 120*n
%e A094216 +1524*C(n,2) +8200*C(n,3) +24860*C(n,4) +47185*C(n,5) +58465*C(n,6)
%e A094216 +47474*C(n,7) +24430*C(n,8) +7245*C(n,9) +945*C(n,10). And then substituting
%e A094216 for the 10th number of such a S1(n+p,n) gives S1(15,10) = 37312275.
%Y A094216 Cf. A000914, A001303, A000915, A053567, A008275, A008276.
%Y A094216 Cf. A000012, A000217, A001147, A000142, A094262.
%Y A094216 Sequence in context: A011416 A086658 A011053 this_sequence A102098 A019731 
               A021363
%Y A094216 Adjacent sequences: A094213 A094214 A094215 this_sequence A094217 A094218 
               A094219
%K A094216 easy,nonn,tabl
%O A094216 1,3
%A A094216 Andre F. Labossiere (boronali(AT)laposte.net), May 27 2004, Feb 21 2007

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