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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). 22
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; text; internal format)
OFFSET

1,3

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.

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

Table of n, a(n) for n=1..43.

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.

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).

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.

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 *)

CROSSREFS

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

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

Sequence in context: A189039 A198815 A011053 * A197495 A102098 A316252

Adjacent sequences: A094213 A094214 A094215 * A094217 A094218 A094219

KEYWORD

easy,nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified December 7 19:19 EST 2022. Contains 358669 sequences. (Running on oeis4.)