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A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.

%I A123021 #23 Feb 16 2025 08:33:02
%S A123021 1,2,-1,3,-2,4,-2,-2,1,5,0,-9,6,-1,6,5,-24,18,-4,7,14,-49,36,-4,-4,1,
%T A123021 8,28,-84,50,20,-30,10,-1,9,48,-126,36,115,-120,45,-6,10,75,-168,-48,
%U A123021 358,-335,120,-6,-6,1,11,110,-198,-264,847,-714,175,84,-63,14
%N A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.
%C A123021 The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A078812(n,j)*x^j*(1 - x)^(n - j).
%H A123021 G. C. Greubel, <a href="/A123021/b123021.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%H A123021 Thomas Koshy, <a href="https://doi.org/10.1002/9781118033067.ch41">Morgan-Voyce Polynomials</a>, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, pp. 480-495.
%H A123021 M. N. S. Swamy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-1/swamy2.pdf">Rising Diagonal Polynomials Associated with Morgan-Voyce Polynomials</a>, The Fibonacci Quarterly Vol. 38 (2000), 61-70.
%H A123021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%F A123021 From _Franck Maminirina Ramaharo_, Oct 09 2018: (Start)
%F A123021 Row n = coefficients in the expansion of (1/sqrt((4 - 3*x)*x))*(((2 - x + sqrt((4 - 3*x)*x))/2)^(n + 1) - ((2 - x - sqrt((4 - 3*x)*x))/2)^(n + 1)).
%F A123021 G.f.: 1/(1 - (2 - x)*y + (1 - x)^2*y^2).
%F A123021 E.g.f.: (1/sqrt((4 - 3*x)*x))*((2 - x + sqrt((4 - 3*x)*x))*exp(y*(2 - x + sqrt((4 - 3*x)*x))/2)/2 - (2 - x - sqrt((4 - 3*x)*x))*exp(y*(2 - x - sqrt((4 - 3*x)*x))/2)/2).
%F A123021 T(n,1) = -A254749(n+1). (End)
%e A123021 Triangle begins:
%e A123021     1;
%e A123021     2,  -1;
%e A123021     3,  -2;
%e A123021     4,  -2,   -2,    1;
%e A123021     5,   0,   -9,    6,  -1;
%e A123021     6,   5,  -24,   18,  -4;
%e A123021     7,  14,  -49,   36,  -4,   -4,   1;
%e A123021     8,  28,  -84,   50,  20,  -30,  10, -1;
%e A123021     9,  48, -126,   36, 115, -120,  45, -6;
%e A123021    10,  75, -168,  -48, 358, -335, 120, -6,  -6,  1;
%e A123021    11, 110, -198, -264, 847, -714, 175, 84, -63, 14, -1;
%e A123021    ... - _Franck Maminirina Ramaharo_, Oct 09 2018
%t A123021 Table[CoefficientList[Sum[Binomial[n+k+1, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten
%o A123021 (Maxima) t(n, k) := binomial(n + k + 1, n - k)$
%o A123021 P(x, n) := expand(sum(t(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
%o A123021 T(n, k) := ratcoef(P(x, n), x, k)$
%o A123021 tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* _Franck Maminirina Ramaharo_, Oct 09 2018 */
%o A123021 (Sage)
%o A123021 def p(n,x): return sum( binomial(n+j+1, n-j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o A123021 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A123021 flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 15 2021
%Y A123021 Cf. A078812, A085478.
%Y A123021 Cf. A122753, A123018, A123019, A123027, A123199, A123202, A123217, A123221.
%K A123021 sign,tabf
%O A123021 0,2
%A A123021 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2006
%E A123021 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 09 2018