OFFSET
0,2
COMMENTS
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).
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
FORMULA
From Franck Maminirina Ramaharo, Oct 09 2018: (Start)
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)).
G.f.: 1/(1 - (2 - x)*y + (1 - x)^2*y^2).
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).
T(n,1) = -A254749(n+1). (End)
EXAMPLE
Triangle begins:
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;
8, 28, -84, 50, 20, -30, 10, -1;
9, 48, -126, 36, 115, -120, 45, -6;
10, 75, -168, -48, 358, -335, 120, -6, -6, 1;
11, 110, -198, -264, 847, -714, 175, 84, -63, 14, -1;
... - Franck Maminirina Ramaharo, Oct 09 2018
MATHEMATICA
Table[CoefficientList[Sum[Binomial[n+k+1, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten
PROG
(Maxima) t(n, k) := binomial(n + k + 1, n - k)$
P(x, n) := expand(sum(t(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 */
(Sage)
def p(n, x): return sum( binomial(n+j+1, n-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
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 24 2006
EXTENSIONS
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 09 2018
STATUS
approved