OFFSET
0,5
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
LINKS
Reinhard Zumkeller, Rows n = 0..125 of table, flattened
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.
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
EXAMPLE
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)
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 23 2003
STATUS
approved