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 A230206 Trapezoid of dot products of row 3 (signs alternating) with sequential 4-tuples read by rows in Pascal's triangle A007318: T(n,k) is the linear combination of the 4-tuples (C(3,0), -C(3,1), C(3,2), -C(3,3)) and (C(n-1,k-3), C(n-1,k-2), C(n-1,k-1), C(n-1,k)), n >= 1, 0 <= k <= n+2. 7
 -1, 3, -3, 1, -1, 2, 0, -2, 1, -1, 1, 2, -2, -1, 1, -1, 0, 3, 0, -3, 0, 1, -1, -1, 3, 3, -3, -3, 1, 1, -1, -2, 2, 6, 0, -6, -2, 2, 1, -1, -3, 0, 8, 6, -6, -8, 0, 3, 1, -1, -4, -3, 8, 14, 0, -14, -8, 3, 4, 1, -1, -5, -7, 5, 22, 14, -14, -22, -5, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The array is trapezoidal rather than triangular because C(n,k) is not uniquely defined for all negative n and negative k. Row sums are 0. Coefficients of (x-1)^3 (x+1)^(n-1) for n > 0. LINKS Dixon J. Jones, Rows n = 1..100 for irregular triangle, flattened Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4. FORMULA T(n,k) = Sum_{i=0..n+m-1} (-1)^(i+m)*C(m,i)*C(n-1,k-i), n >= 1, with T(n,0) = (-1)^m and m=3. EXAMPLE Trapezoid begins -1, 3, -3, 1; -1, 2, 0, -2, 1; -1, 1, 2, -2, -1, 1; -1, 0, 3, 0, -3, 0, 1; -1, -1, 3, 3, -3, -3, 1, 1; -1, -2, 2, 6, 0, -6, -2, 2, 1; -1, -3, 0, 8, 6, -6, -8, 0, 3, 1; MATHEMATICA Flatten[Table[CoefficientList[(x - 1)^3 (x + 1)^n, x], {n, 0, 7}]] (* T. D. Noe, Oct 25 2013 *) m=3; Table[If[k == 0, (-1)^m, Sum[(-1)^(j+m)*Binomial[m, j]*Binomial[n-1, k-j], {j, 0, n+m-1}]], {n, 1, 10}, {k, 0, n+m-1}]//Flatten (* G. C. Greubel, Nov 29 2018 *) PROG (PARI) m=3; for(n=1, 10, for(k=0, n+m-1, print1(if(k==0, (-1)^m, sum(j=0, n+m-1, (-1)^(j+m)*binomial(m, j)*binomial(n-1, k-j))), ", "))) \\ G. C. Greubel, Nov 29 2018 (Magma) m:=3; [[k le 0 select (-1 )^m else (&+[(-1)^(j+m)* Binomial(m, j) *Binomial(n-1, k-j): j in [0..(n+m-1)]]): k in [0..(n+m-1)]]: n in [1..10]]; // G. C. Greubel, Nov 29 2018 (Sage) m=3; [[sum((-1)^(j+m)*binomial(m, j)*binomial(n-1, k-j) for j in range(n+m)) for k in range(n+m)] for n in (1..10)] # G. C. Greubel, Nov 29 2018 CROSSREFS Using row j of the alternating Pascal triangle as generator: A007318 (j=0), A008482 and A112467 (j=1 after the first term in each), A182533 (j=2 after the first two rows), A230207-A230212 (j=4 to j=9). Sequence in context: A072917 A319861 A114266 * A285117 A135910 A255916 Adjacent sequences: A230203 A230204 A230205 * A230207 A230208 A230209 KEYWORD easy,sign,tabf AUTHOR Dixon J. Jones, Oct 11 2013 STATUS approved

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Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)