A123562 Pascal-(1,-3,1) array, read by antidiagonals.

1, 1, 1, 1, -1, 1, 1, -3, -3, 1, 1, -5, -3, -5, 1, 1, -7, 1, 1, -7, 1, 1, -9, 9, 11, 9, -9, 1, 1, -11, 21, 17, 17, 21, -11, 1, 1, -13, 37, 11, 1, 11, 37, -13, 1, 1, -15, 57, -15, -39, -39, -15, 57, -15, 1, 1, -17, 81, -69, -87, -81, -87, -69, 81, -17, 1

Offset: 0

Author

Philippe Deléham, Nov 12 2006

Keywords

• sign
• tabl
• easy

Comments

Riordan array (1/(1-x), x*(1-3x)/(1-x)).

Examples

			Triangle begins:
  1;
  1,   1;
  1,  -1,   1;
  1,  -3,  -3,   1;
  1,  -5,  -3,  -5,   1;
  1,  -7,   1,   1,  -7,   1;
  1,  -9,   9,  11,   9,  -9,  1;
  1, -11,  21,  17,  17,  21, -11,   1;
  1, -13,  37,  11,   1,  11,  37, -13,   1;
		

Links

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

Crossrefs

Cf. Pascal (1,m,1) array: A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Programs

Mathematica

T[n_, k_] := Sum[Binomial[n - j, k]*Binomial[k, j]*(-3)^j, {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 15 2017 *)

PARI

for(n=0,10, for(k=0,n, print1(sum(j=0,n, binomial(n-j,k)*  binomial(k,j)*(-3)^j), ", "))) \\ G. C. Greubel, Oct 15 2017

Formula

Sum_{k=0..n} T(n,k) = A088137(n+1).

T(n,k) = T(n-1,k-1) + T(n-1,k) - 3*T(n-2,k-1), n>0.

From Paul Barry, Jan 24 2011: (Start)

T(n,k) = Sum_{j=0..n} binomial(n-j,k)*binomial(k,j)*(-3)^j.

T(n,k) = [k<=n]*Hypergeometric2F1(-k,k-n,1,-2). (End)

E.g.f. for the n-th subdiagonal: exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(-2*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 - 4*x + 4*x^2/2) = 1 - 3*x - 3*x^2/2! + x^3/3! + 9*x^4/4! + 21*x^5/5! + .... - Peter Bala, Mar 05 2017