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A110522 Riordan array (1/(1+x), x(1-2x)/(1+x)^2). 3
1, -1, 1, 1, -5, 1, -1, 12, -9, 1, 1, -22, 39, -13, 1, -1, 35, -115, 82, -17, 1, 1, -51, 270, -344, 141, -21, 1, -1, 70, -546, 1106, -773, 216, -25, 1, 1, -92, 994, -2954, 3199, -1466, 307, -29, 1, -1, 117, -1674, 6888, -10791, 7461, -2487, 414, -33, 1, 1, -145, 2655, -14484, 31179, -30645, 15060, -3900, 537, -37, 1, -1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Inverse of A110519. Row sums are A110523. Diagonal sums are A110524. Product of inverse binomial transform matrix (1/(1+x), x/(1+x)) and (1, x(1-3x)) (A110517).

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, J. Int. Seq. 16 (2013) #13.5.4

FORMULA

Number triangle T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).

T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k)*2^i.

T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - 2*T(n-2,k-1), T(0,0) = 1, T(1,0) = -1, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014

EXAMPLE

Rows begin

   1;

  -1,    1;

   1,   -5,    1;

  -1,   12,   -9,    1;

   1,  -22,   39,  -13,    1;

  -1,   35, -115,   82,  -17,    1;

MATHEMATICA

T[0, 0] := 1; T[n_, k_] := Sum[(-1)^(n - j)*(-3)^(j - k)*Binomial[k, j - k]*Binomial[n, j], {j, 0, n}]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 30 2017 *)

PROG

(PARI) concat([1], for(n=1, 20, for(k=0, n, print1(sum(j=0, n, (-1)^(n-j)*(-3)^(j-k)*binomial(n, j)*binomial(k, j-k)), ", ")))) \\ G. C. Greubel, Aug 30 2017

CROSSREFS

Sequence in context: A146954 A174949 A174861 * A146987 A297915 A298508

Adjacent sequences:  A110519 A110520 A110521 * A110523 A110524 A110525

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Jul 24 2005

STATUS

approved

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Last modified November 29 19:26 EST 2020. Contains 338769 sequences. (Running on oeis4.)