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A123019 Triangle of coefficients of (1 - x)^n*b(x/(1 - x),n), where b(x,n) is the Morgan-Voyce polynomial related to A085478.

%I A123019 #34 Feb 16 2025 08:33:02
%S A123019 1,1,1,1,-1,1,3,-4,1,1,6,-9,3,1,10,-15,3,3,-1,1,15,-20,-6,18,-8,1,1,
%T A123019 21,-21,-35,60,-30,5,1,28,-14,-98,145,-70,5,5,-1,1,36,6,-210,279,-100,
%U A123019 -45,45,-12,1,1,45,45,-384,441,-21,-280,210,-63,7,1,55,110
%N A123019 Triangle of coefficients of (1 - x)^n*b(x/(1 - x),n), where b(x,n) is the Morgan-Voyce polynomial related to A085478.
%C A123019 The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A085478(n,j)*x^j*(1 - x)^(n - j).
%H A123019 G. C. Greubel, <a href="/A123019/b123019.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%H A123019 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 A123019 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 A123019 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%F A123019 G.f.: (1 - (1 - x)*y)/(1 + (x - 2)*y + (x - 1)^2*y^2). - _Vladeta Jovovic_, Dec 14 2009
%F A123019 From _Franck Maminirina Ramaharo_, Oct 10 2018: (Start)
%F A123019 Row n = coefficients in the expansion of (1/(2*sqrt((4 - 3*x)*x)))*((sqrt((4 - 3*x)*x) + x)*((2 - x + sqrt((4 - 3*x)*x))/2)^n + (sqrt((4 - 3*x)*x) - x)*((2 - x - sqrt((4 - 3*x)*x))/2)^n).
%F A123019 E.g.f.: (1/(2*sqrt((4 - 3*x)*x)))*((sqrt((4 - 3*x)*x) + x)*exp(y*(2 - x + sqrt((4 - 3*x)*x))/2) + (sqrt((4 - 3*x)*x) - x)*exp(y*(2 - x - sqrt((4 - 3*x)*x))/2)).
%F A123019 T(n,1) = A000217(n-1). (End)
%e A123019 Triangle begins:
%e A123019     1;
%e A123019     1;
%e A123019     1,  1,  -1;
%e A123019     1,  3,  -4,    1;
%e A123019     1,  6,  -9,    3;
%e A123019     1, 10, -15,    3,   3,   -1;
%e A123019     1, 15, -20,   -6,  18,   -8,    1;
%e A123019     1, 21, -21,  -35,  60,  -30,    5;
%e A123019     1, 28, -14,  -98, 145,  -70,    5,   5,   -1;
%e A123019     1, 36,   6, -210, 279, -100,  -45,  45,  -12, 1;
%e A123019     1, 45,  45, -384, 441,  -21, -280, 210,  -63, 7;
%e A123019     1, 55, 110, -627, 561,  385, -973, 665, -189, 7, 7, -1;
%e A123019     ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 09 2018
%t A123019 Table[CoefficientList[Sum[Binomial[n+k, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten
%o A123019 (Maxima) A085478(n, k) := binomial(n + k, 2*k)$
%o A123019 P(x, n) := expand(sum(A085478(n, j)*x^j*(1 - x)^(n - j),j,0,n))$
%o A123019 T(n, k) := ratcoef(P(x, n), x, k)$
%o A123019 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 A123019 (Sage)
%o A123019 def p(n,x): return sum( binomial(n+j, 2*j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o A123019 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A123019 flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 15 2021
%Y A123019 Cf. A078812, A085478.
%Y A123019 Cf. A122753, A123018, A123021, A123027, A123199, A123202, A123217, A123221.
%K A123019 sign,tabf
%O A123019 0,7
%A A123019 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2006
%E A123019 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 09 2018