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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 G. C. Greubel, Rows n = 1..50 of the triangle, flattened 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. Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128. Sequence in context: A093430 A074659 A131251 * A225281 A057145 A134394 Adjacent sequences: A144397 A144398 A144399 * A144401 A144402 A144403 KEYWORD tabl,sign AUTHOR Roger L. Bagula and Gary W. Adamson, Oct 03 2008 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.)