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A230212 Trapezoid of dot products of row 9 (signs alternating) with sequential 10-tuples read by rows in Pascal's triangle A007318: T(n,k) is the linear combination of the 10-tuples (C(9,0), -C(9,1), ..., C(9,8), -C(9,9)) and (C(n-1,k-9), C(n-1,k-8), ..., C(n-1,k)), n >= 1, 0 <= k <= n+8. 7
-1, 9, -36, 84, -126, 126, -84, 36, -9, 1, -1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, -1, 7, -19, 21, 6, -42, 42, -6, -21, 19, -7, 1, -1, 6, -12, 2, 27, -36, 0, 36, -27, -2, 12, -6, 1, -1, 5, -6, -10, 29, -9, -36, 36, 9, -29, 10, 6, -5, 1, -1, 4, -1, -16 (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)^9 (x+1)^(n-1), n > 0.

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)*C(m,i)*C(n-1,k-i))], n>=1, with T(n,0) = (-1)^m and m=9.

EXAMPLE

Trapezoid begins:

-1, 9, -36, 84, -126, 126, -84, 36, -9, 1;

-1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1;

-1, 7, -19, 21, 6, -42, 42, -6, -21, 19, -7, 1;

-1, 6, -12, 2, 27, -36, 0, 36, -27, -2, 12, -6, 1;

-1, 5, -6, -10, 29, -9, -36, 36, 9, -29, 10, 6, -5, 1;

-1, 4, -1, -16, 19, 20, -45, 0, 45, -20, -19, 16, 1, -4, 1;

-1, 3, 3, -17, 3, 39, -25, -45, 45, 25, -39, -3, 17, -3, -3, 1;

etc.

MATHEMATICA

Flatten[Table[CoefficientList[(x - 1)^9 (x + 1)^n, x], {n, 0, 7}]] (* T. D. Noe, Oct 25 2013 *)

m=9; 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=9; 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:=9; [[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=9; [[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-A230211 (j=3 to j=8).

Sequence in context: A001487 A175673 A285198 * A229394 A173682 A009522

Adjacent sequences: A230209 A230210 A230211 * A230213 A230214 A230215

KEYWORD

easy,sign,tabf

AUTHOR

Dixon J. Jones, Oct 12 2013

STATUS

approved

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Last modified December 5 15:24 EST 2022. Contains 358588 sequences. (Running on oeis4.)