A081577 Pascal-(1,2,1) array read by antidiagonals.
1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 22, 10, 1, 1, 13, 46, 46, 13, 1, 1, 16, 79, 136, 79, 16, 1, 1, 19, 121, 307, 307, 121, 19, 1, 1, 22, 172, 586, 886, 586, 172, 22, 1, 1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1, 1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1
Offset: 0
Examples
Square array begins as: 1, 1, 1, 1, 1, ... A000012; 1, 4, 7, 10, 13, ... A016777; 1, 7, 22, 46, 79, ... A038764; 1, 10, 46, 136, 307, ... A081583; 1, 13, 79, 307, 886, ... A081584; From _Roger L. Bagula_, Dec 09 2008: (Start) As a triangle this begins: 1; 1, 1; 1, 4, 1; 1, 7, 7, 1; 1, 10, 22, 10, 1; 1, 13, 46, 46, 13, 1; 1, 16, 79, 136, 79, 16, 1; 1, 19, 121, 307, 307, 121, 19, 1; 1, 22, 172, 586, 886, 586, 172, 22, 1; 1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1; 1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1; (End)
Links
- Reinhard Zumkeller, Rows n = 0..125 of table, flattened
- Peter Bala, A note on the diagonals of a proper Riordan Array
- Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
- Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
- Shanghua Zheng, Li Guo, and Huizhen Qiu, Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras, arXiv:2401.11363 [math.RA], 2024. See pp. 44-45.
Crossrefs
Programs
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Haskell
a081577 n k = a081577_tabl !! n !! k a081577_row n = a081577_tabl !! n a081577_tabl = map fst $ iterate (\(us, vs) -> (vs, zipWith (+) (map (* 2) ([0] ++ us ++ [0])) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1]) -- Reinhard Zumkeller, Mar 16 2014 -
Magma
A081577:= func< n,k | (&+[Binomial(k,j)*Binomial(n-j,k)*2^j: j in [0..n-k]]) >; [A081577(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 25 2021
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Mathematica
a[0]={1}; a[1]={1, 1}; a[n_]:= a[n]= 2*Join[{0}, a[n-2], {0}] + Join[{0}, a[n-1]] + Join[a[n-1], {0}]; Table[a[n], {n,0,10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *) Table[Hypergeometric2F1[-k, k-n, 1, 3], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *) -
Sage
flatten([[hypergeometric([-k, k-n], [1], 3).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021
Formula
Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+2*x)^k/(1-x)^(k+1).
G.f.: 1/(1-x-y-2*x*y). - Ralf Stephan, Apr 28 2004
T(n,k) = Sum_{j=0..n} binomial(k,j-k)*binomial(n+k-j,k)*2^(j-k). - Paul Barry, Oct 23 2006
a(n) = 2*{0, a(n-2), 0} + {0, a(n-1)} + {a(n-1), 0}. - Roger L. Bagula, Dec 09 2008
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 3). - Jean-François Alcover, May 24 2013
The e.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(3*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 6*x + 9*x^2/2) = 1 + 7*x + 22*x^2/2! + 46*x^3/3! + 79*x^4/4! + 121*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A002605(n). - G. C. Greubel, May 25 2021
Comments