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A259456
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Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.
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4
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1, 2, 3, 6, 20, 15, 24, 130, 210, 105, 120, 924, 2380, 2520, 945, 720, 7308, 26432, 44100, 34650, 10395, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025, 362880, 6636960, 47324376, 177331440, 389449060, 520059540, 416215800
(list;
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text;
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OFFSET
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0,2
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REFERENCES
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L. Comtet, Advanced Combinatorics (1974), Chapter VI, page 256.
DJ Jeffrey, GA Kalugin, N Murdoch, Lagrange inversion and Lambert W, Preprint 2015; http://www.apmaths.uwo.ca/~djeffrey/Offprints/JeffreySYNASC2015paper17.pdf
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.
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LINKS
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FORMULA
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T(n,k) = (n-k-1)*( T(n-1,k-1)+T(n-1,k) ), n>=1, 1<=k<=n. [Berg, Eq. 6]
The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016
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EXAMPLE
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Triangle begins:
1,
2,3,
6,20,15,
24,130,210,105,
120,924,2380,2520,945,
...
For k=4 and j=2 in Knuth's equation, |S1(4,4-2)| = |S1(4,2)| = |A008275(4,2)| = 11 = p_{2,1}*C(4,3) +p_{2,2}*C(4,4) = 2*4+3*1. - R. J. Mathar, Jul 16 2015
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MAPLE
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option remember;
if k < 1 or k > n then
0 ;
elif n = 1 then
1;
else
procname(n-1, k-1)+procname(n-1, k);
%*(n+k-1) ;
end if;
end proc:
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[k < 1 || k > n, 0, If[n == 1, 1, (T[n-1, k-1] + T[n-1, k])(n+k-1)]];
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CROSSREFS
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Cf. This is a row reversed and unsigned version of A111999.
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KEYWORD
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AUTHOR
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STATUS
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approved
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