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A122753
Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial.
14
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, 0, 1, 502, 5278, -16981, 5335, 30871
OFFSET
0,12
COMMENTS
The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A048993(n,j)*x^j*(1 - x)^(n - j), where A048993 is the triangle of Stirling numbers of second kind.
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
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
KEYWORD
sign,tabf
AUTHOR
EXTENSIONS
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018
STATUS
approved