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A094216
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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).
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22
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1, 1, 2, 7, 8, 3, 6, 38, 93, 111, 65, 15, 24, 226, 874, 1821, 2224, 1600, 630, 105, 120, 1524, 8200, 24860, 47185, 58465, 47474, 24430, 7245, 945, 720, 11628, 81080, 326712, 852690, 1522375, 1905168, 1676325, 1018682, 407925, 97020, 10395, 5040
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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.
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REFERENCES
| 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.
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LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Francis L. Miksa (1901-1975), Stirling numbers of the first kind, "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]).
Dragoslav S. Mitrinovic (1908-1995), Sur les nombres de Stirling de premiere espece et les polynomes de Stirling, 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.
John J. O'Connor and Edmund F. Robertson, James Stirling (1692-1770), (September 1998).
Eric Weisstein's World of Mathematics, Stirling numbers of the first kind.
Stephen Wolfram, Wolfram Research, Mathematica 5.2, webMathematica 2.
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FORMULA
| a(1,k) = k!
...
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)
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)
a(2*k-3,k) = a(2*k,k) * (1250*k^4-4225*k^3+5023*k^2-2600*k+528) / (1620*k-810)
a(2*k-2,k) = a(2*k,k) * (50*k^3-93*k^2+55*k-12) / (36*k-18)
a(2*k-1,k) = a(2*k,k) * (5*k-2) / 3
a(2*k,k) = (2*k)! / (k!*2^k).
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EXAMPLE
| 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.
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CROSSREFS
| Cf. A000914, A001303, A000915, A053567, A008275, A008276.
Cf. A000012, A000217, A001147, A000142, A094262.
Sequence in context: A189039 A198815 A011053 * A197495 A102098 A202355
Adjacent sequences: A094213 A094214 A094215 * A094217 A094218 A094219
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Andre F. Labossiere (boronali(AT)laposte.net), May 27 2004, Feb 21 2007
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