cp's OEIS Frontend

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).

%I A144400 #16 Jul 15 2021 01:56:13
%S A144400 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 A144400 -63,70,-24,8,-28,56,0,-168,280,-192,49,9,-36,84,0,-378,840,-864,441,
%U A144400 -89,10,-45,120,0,-756,2100,-2880,2205,-890,145,11,-55,165,0
%N 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).
%H A144400 G. C. Greubel, <a href="/A144400/b144400.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144400 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 A144400 Triangle begins:
%e A144400     1;
%e A144400     2,  -1;
%e A144400     3,  -3,   1;
%e A144400     4,  -6,   4, 0;
%e A144400     5, -10,  10, 0,   -3;
%e A144400     6, -15,  20, 0,  -18,   10;
%e A144400     7, -21,  35, 0,  -63,   70,   -24;
%e A144400     8, -28,  56, 0, -168,  280,  -192,   49;
%e A144400     9, -36,  84, 0, -378,  840,  -864,  441,  -89;
%e A144400    10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
%e A144400      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 22 2018
%t A144400 a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
%t A144400 p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
%t A144400 Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten
%o A144400 (Sage)
%o A144400 @CachedFunction
%o A144400 def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
%o A144400 def p(n,x): return sum( binomial(n,j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
%o A144400 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A144400 [T(n) for n in (1..12)] # _G. C. Greubel_, Jul 14 2021
%Y A144400 Row sums: subsequence of A000931, A078027, A182097.
%Y A144400 Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.
%K A144400 tabl,sign
%O A144400 1,2
%A A144400 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 03 2008
%E A144400 Edited, and new name by _Franck Maminirina Ramaharo_, Oct 22 2018