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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.

%I A122753 #30 Feb 16 2025 08:33:02
%S A122753 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,
%T A122753 1,57,1,-244,281,-104,9,0,1,120,225,-1259,1401,-454,-83,50,0,1,247,
%U A122753 1268,-5081,4621,911,-3422,1723,-267,0,1,502,5278,-16981,5335,30871
%N A122753 Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial.
%C A122753 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.
%D A122753 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.
%H A122753 G. C. Greubel, <a href="/A122753/b122753.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%H A122753 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A122753 Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%H A122753 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%H A122753 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling Number of the Second Kind</a>
%F A122753 From _Franck Maminirina Ramaharo_, Oct 10 2018: (Start)
%F A122753 E.g.f.: exp((x/(1 - x))*(exp((1 - x)*y) - 1)).
%F A122753 T(n,1) = A000295(n-1). (End)
%e A122753 Triangle begins:
%e A122753     1;
%e A122753     0, 1;
%e A122753     0, 1;
%e A122753     0, 1,   1,   -1;
%e A122753     0, 1,   4,   -5,     1;
%e A122753     0, 1,  11,  -14,     1,    2;
%e A122753     0, 1,  26,  -24,   -29,   36,   -9;
%e A122753     0, 1,  57,    1,  -244,  281, -104,     9;
%e A122753     0, 1, 120,  225, -1259, 1401, -454,   -83,   50;
%e A122753     0, 1, 247, 1268, -5081, 4621,  911, -3422, 1723, -267;
%e A122753     ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 10 2018
%t A122753 Table[CoefficientList[Sum[StirlingS2[m, n]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten
%o A122753 (Maxima)
%o A122753 P(x, n) := expand(sum(stirling2(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
%o A122753 T(n, k) := ratcoef(P(x, n), x, k)$
%o A122753 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 */
%o A122753 (Sage)
%o A122753 def p(n,x): return sum( stirling_number2(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o A122753 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A122753 flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 15 2021
%Y A122753 Cf. A000110, A048993, A088996, A122610.
%Y A122753 Cf. A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221.
%K A122753 sign,tabf
%O A122753 0,12
%A A122753 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 21 2006
%E A122753 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 10 2018