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

0,7

Comments

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).

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

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)

Example

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

Mathematica

Table[CoefficientList[Sum[Binomial[n+k, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten

Prog

(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 */
(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

Crossrefs

Cf. A078812, A085478.

Cf. A122753, A123018, A123021, A123027, A123199, A123202, A123217, A123221.

Sequence in context: A001086 A105283 A347178 * A226063 A326748 A204999

Adjacent sequences: A123016 A123017 A123018 * A123020 A123021 A123022

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