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%I #21 Jul 16 2021 01:33:54
%S 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 8,28,-84,50,20,-30,10,-1,9,48,-126,36,115,-120,45,-6,10,75,-168,-48,
%U 358,-335,120,-6,-6,1,11,110,-198,-264,847,-714,175,84,-63,14
%N 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 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 G. C. Greubel, <a href="/A123021/b123021.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%H 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 M. N. S. Swamy, <a href="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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%F From _Franck Maminirina Ramaharo_, Oct 09 2018: (Start)
%F 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 G.f.: 1/(1 - (2 - x)*y + (1 - x)^2*y^2).
%F 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 T(n,1) = -A254749(n+1). (End)
%e Triangle begins:
%e 1;
%e 2, -1;
%e 3, -2;
%e 4, -2, -2, 1;
%e 5, 0, -9, 6, -1;
%e 6, 5, -24, 18, -4;
%e 7, 14, -49, 36, -4, -4, 1;
%e 8, 28, -84, 50, 20, -30, 10, -1;
%e 9, 48, -126, 36, 115, -120, 45, -6;
%e 10, 75, -168, -48, 358, -335, 120, -6, -6, 1;
%e 11, 110, -198, -264, 847, -714, 175, 84, -63, 14, -1;
%e ... - _Franck Maminirina Ramaharo_, Oct 09 2018
%t Table[CoefficientList[Sum[Binomial[n+k+1, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten
%o (Maxima) t(n, k) := binomial(n + k + 1, n - k)$
%o P(x, n) := expand(sum(t(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
%o T(n, k) := ratcoef(P(x, n), x, k)$
%o 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 (Sage)
%o def p(n,x): return sum( binomial(n+j+1, n-j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 15 2021
%Y Cf. A078812, A085478.
%Y Cf. A122753, A123018, A123019, A123027, A123199, A123202, A123217, A123221.
%K sign,tabf
%O 0,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2006
%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 09 2018
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