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A123562 Pascal-(1,-3,1) array, read by antidiagonals. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Riordan array (1/(1-x), x*(1-3x)/(1-x)).
LINKS
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
EXAMPLE
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;
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 *)
PROG
(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
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).
Sequence in context: A133332 A362895 A179680 * A046218 A046221 A220837
KEYWORD
sign,tabl,easy
AUTHOR
Philippe Deléham, Nov 12 2006
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)