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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

%I #16 Jul 15 2021 01:56:13

%S 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,

%T -63,70,-24,8,-28,56,0,-168,280,-192,49,9,-36,84,0,-378,840,-864,441,

%U -89,10,-45,120,0,-756,2100,-2880,2205,-890,145,11,-55,165,0

%N 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).

%H G. C. Greubel, <a href="/A144400/b144400.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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

%e Triangle begins:

%e 1;

%e 2, -1;

%e 3, -3, 1;

%e 4, -6, 4, 0;

%e 5, -10, 10, 0, -3;

%e 6, -15, 20, 0, -18, 10;

%e 7, -21, 35, 0, -63, 70, -24;

%e 8, -28, 56, 0, -168, 280, -192, 49;

%e 9, -36, 84, 0, -378, 840, -864, 441, -89;

%e 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;

%e ... reformatted. - _Franck Maminirina Ramaharo_, Oct 22 2018

%t a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];

%t p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];

%t Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten

%o (Sage)

%o @CachedFunction

%o def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)

%o def p(n,x): return sum( binomial(n,j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )

%o def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)

%o [T(n) for n in (1..12)] # _G. C. Greubel_, Jul 14 2021

%Y Row sums: subsequence of A000931, A078027, A182097.

%Y Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.

%K tabl,sign

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 03 2008

%E Edited, and new name by _Franck Maminirina Ramaharo_, Oct 22 2018