A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).
1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -1, -10, -44, -110, -165, -132, 0, 132, 165, 110
Offset: 0
Examples
Triangle starts:
1;
-1, 1;
-1, 0, 1;
-1, -1, 1, 1;
-1, -2, 0, 2, 1;
-1, -3, -2, 2, 3, 1;
-1, -4, -5, 0, 5, 4, 1;
-1, -5, -9, -5, 5, 9, 5, 1;
-1, -6, -14, -14, 0, 14, 14, 6, 1;
-1, -7, -20, -28, -14, 14, 28, 20, 7, 1;
-1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1;
-1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins:
1, 1,
-2, -1, 1,
2, 0, -1, 1,
-2, 0, 0, -1, 1,
2, 0, 0, 0, -1, 1,
-2, 0, 0, 0, 0, -1, 1,
2, 0, 0, 0, 0, 0, -1, 1
... (End)
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
- Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Restricting Dyck Paths and 312-avoiding Permutations, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.
- Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
- Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
- D. Foata and G.-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
- Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]
Crossrefs
Programs
-
Magma
[n eq 0 select 1 else (2*k-n)*Binomial(n,k)/n: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 04 2019
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Maple
seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n,k)/n), k=0..n), n=0..10); # G. C. Greubel, Dec 04 2019
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Mathematica
T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019 *) -
PARI
T(n, k) = if(n==0, 1, (2*k-n)*binomial(n,k)/n ); \\ G. C. Greubel, Dec 04 2019
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Sage
def T(n, k): if (n==0): return 1 else: return (2*k-n)*binomial(n,k)/n [[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019
Formula
Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).
Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - Philippe Deléham, Oct 03 2005
T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019
T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - Philippe Deléham, Nov 01 2011
G.f.: (1-2x)/(1-(1+y)*x). - Philippe Deléham, Dec 15 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - Philippe Deléham, Dec 15 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - G. C. Greubel, Dec 04 2019
Comments