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 A230209 Trapezoid of dot products of row 6 (signs alternating) with sequential 7-tuples read by rows in Pascal's triangle A007318: T(n,k) is the linear combination of the 7-tuples (C(6,0), -C(6,1), ..., -C(6,5), C(6,6)) and (C(n-1,k-6), C(n-1,k-5), ..., C(n-1,k)), n >= 1, 0 <= k <= n+5. 3
 1, -6, 15, -20, 15, -6, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -4, 4, 4, -10, 4, 4, -4, 1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -2, -3, 8, 2, -12, 2, 8, -3, -2, 1, 1, -1, -5, 5, 10, -10, -10, 10, 5, -5, -1, 1, 1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1, 1, 1, -6 (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)^6 (x+1)^(n-1). LINKS G. C. Greubel, Rows n=1..50 of trapezoid, flattened Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, 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=6. EXAMPLE Trapezoid begins: 1, -6, 15, -20, 15, -6, 1; 1, -5, 9, -5, -5, 9, -5, 1; 1, -4, 4, 4, -10, 4, 4, -4, 1; 1, -3, 0, 8, -6, -6, 8, 0, -3, 1; 1, -2, -3, 8, 2, -12, 2, 8, -3, -2, 1; 1, -1, -5, 5, 10, -10, -10, 10, 5, -5, -1, 1; 1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1; etc. MATHEMATICA Flatten[Table[CoefficientList[(x - 1)^6 (x + 1)^n, x], {n, 0, 7}]] (* T. D. Noe, Oct 25 2013 *) m=6; 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 28 2018 *) PROG (PARI) m=6; 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 28 2018 (Magma) m:=6; [[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 28 2018 (Sage) m=6; [[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 28 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), A230206-A230208 (j=3 to j=5), A230210-A230212 (j=7 to j=9). Sequence in context: A217480 A335417 A173680 * A277951 A176849 A087110 Adjacent sequences: A230206 A230207 A230208 * A230210 A230211 A230212 KEYWORD easy,sign,tabf AUTHOR Dixon J. Jones, Oct 12 2013 STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)