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A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)*binomial(n, j)*x^(j - 1)*(1 - x)^(n - j). 4
1, 2, -1, 3, -3, 1, 4, -6, 4, 0, 5, -10, 10, 0, -3, 6, -15, 20, 0, -18, 10, 7, -21, 35, 0, -63, 70, -24, 8, -28, 56, 0, -168, 280, -192, 49, 9, -36, 84, 0, -378, 840, -864, 441, -89, 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145, 11, -55, 165, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
G.f.: (y - (1 - 2*x)*y^2)/(1 - 3*(1 - x)*y + (3 - 6*x + 2*x^2)*y^2 - (1 - 3*x + 2*x^2 + x^3)*y^3). - Franck Maminirina Ramaharo, Oct 22 2018
EXAMPLE
Triangle begins:
1;
2, -1;
3, -3, 1;
4, -6, 4, 0;
5, -10, 10, 0, -3;
6, -15, 20, 0, -18, 10;
7, -21, 35, 0, -63, 70, -24;
8, -28, 56, 0, -168, 280, -192, 49;
9, -36, 84, 0, -378, 840, -864, 441, -89;
10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
... reformatted. - Franck Maminirina Ramaharo, Oct 22 2018
MATHEMATICA
a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten
PROG
(Sage)
@CachedFunction
def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
def p(n, x): return sum( binomial(n, j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021
CROSSREFS
Row sums: subsequence of A000931, A078027, A182097.
Sequence in context: A093430 A074659 A131251 * A225281 A057145 A134394
KEYWORD
tabl,sign
AUTHOR
EXTENSIONS
Edited, and new name by Franck Maminirina Ramaharo, Oct 22 2018
STATUS
approved

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Last modified March 1 17:48 EST 2024. Contains 370442 sequences. (Running on oeis4.)