A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.

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

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

Cf. A078812, A085478.

Cf. A122753, A123018, A123019, A123027, A123199, A123202, A123217, A123221.

Sequence in context: A373938 A217669 A353420 * A317056 A346699 A360679

Adjacent sequences: A123018 A123019 A123020 * A123022 A123023 A123024

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