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

Author

Paul Barry, Mar 23 2003

Keywords

• easy
• nonn
• tabl

Comments

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016777, A038764, A081583, A081584. Coefficients of the row polynomials in the Newton basis are given by A013610.

As a number triangle, this is the Riordan array (1/(1-x), x(1+2x)/(1-x)). It has row sums A002605 and diagonal sums A077947. - Paul Barry, Jan 24 2005

All entries are == 1 mod 3. - Roger L. Bagula, Oct 04 2008

Row sums are A002605. - Roger L. Bagula, Dec 09 2008

As a number triangle T, T(2n,n)=A069835(n). - Philippe Deléham, Jan 10 2014

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

Cf. A002605, A081578, A081579, A081580.

Cf. Pascal-(1,a,1) array: A123562 (a=-3), A098593 (=-2), A000012 (a=-1), A007318 (a=0), A008288 (a=1), A081577(a=2), A081578 (a=3), A081579 (a=4), A081580 (a=5), A081581 (a=6), A081582 (a=7), A143683(a=8). [From Roger L. Bagula, Dec 09 2008], Philippe Deléham, Jan 10 2014, Mar 16 2014.

Programs

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

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