A094216 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).

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

Offset: 1

Author

André F. Labossière, May 27 2004, Feb 21 2007

Keywords

• easy
• nonn
• tabl

Comments

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.

Examples

			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.
		

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.

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 première espèce et les polynômes de Stirling, 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.
• 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.

Crossrefs

Cf. A000914, A001303, A000915, A053567, A008275, A008276.

Cf. A000012, A000217, A001147, A000142, A094262.

Programs

Mathematica

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];
Array[row, 7] // Flatten (* Jean-François Alcover, Nov 14 2019 *)

Formula

a(1,k) = k!

...

a(2*k-5,k) = a(2*k,k) * (175000*k^8 -2117500*k^7 [phone]*k^6 -30743377*k^5 [phone]*k^4 -55386931*k^3 [phone]*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).