OFFSET
0,12
COMMENTS
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 824-825.
LINKS
G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
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].
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind
FORMULA
From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
E.g.f.: exp((x/(1 - x))*(exp((1 - x)*y) - 1)).
T(n,1) = A000295(n-1). (End)
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1;
0, 1, 1, -1;
0, 1, 4, -5, 1;
0, 1, 11, -14, 1, 2;
0, 1, 26, -24, -29, 36, -9;
0, 1, 57, 1, -244, 281, -104, 9;
0, 1, 120, 225, -1259, 1401, -454, -83, 50;
0, 1, 247, 1268, -5081, 4621, 911, -3422, 1723, -267;
... reformatted and extended. - Franck Maminirina Ramaharo, Oct 10 2018
MATHEMATICA
Table[CoefficientList[Sum[StirlingS2[m, n]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten
PROG
(Maxima)
P(x, n) := expand(sum(stirling2(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */
(Sage)
def p(n, x): return sum( stirling_number2(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 21 2006
EXTENSIONS
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018
STATUS
approved