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
Examples
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
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+1, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten -
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 */
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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
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)
Extensions
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 09 2018
Comments