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.
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, 21, -21, -35, 60, -30, 5, 1, 28, -14, -98, 145, -70, 5, 5, -1, 1, 36, 6, -210, 279, -100, -45, 45, -12, 1, 1, 45, 45, -384, 441, -21, -280, 210, -63, 7, 1, 55, 110
Offset: 0
Examples
Triangle begins:
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, 21, -21, -35, 60, -30, 5;
1, 28, -14, -98, 145, -70, 5, 5, -1;
1, 36, 6, -210, 279, -100, -45, 45, -12, 1;
1, 45, 45, -384, 441, -21, -280, 210, -63, 7;
1, 55, 110, -627, 561, 385, -973, 665, -189, 7, 7, -1;
... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 09 2018
Links
- G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
- Thomas Koshy, Morgan-Voyce Polynomials, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, pp. 480-495.
- M. N. S. Swamy, Rising Diagonal Polynomials Associated with Morgan-Voyce Polynomials, The Fibonacci Quarterly Vol. 38 (2000), 61-70.
- Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Crossrefs
Programs
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Mathematica
Table[CoefficientList[Sum[Binomial[n+k, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten -
Maxima
A085478(n, k) := binomial(n + k, 2*k)$ P(x, n) := expand(sum(A085478(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 09 2018 */
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Sage
def p(n,x): return sum( binomial(n+j, 2*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
Formula
G.f.: (1 - (1 - x)*y)/(1 + (x - 2)*y + (x - 1)^2*y^2). - Vladeta Jovovic, Dec 14 2009
From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
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).
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)).
T(n,1) = A000217(n-1). (End)
Extensions
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 09 2018
Comments