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A081577 Pascal-(1,2,1) array read by antidiagonals. 29
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 (list; table; graph; refs; listen; history; text; internal format)
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

P. 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.

FORMULA

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1)+2T(n-1, k-1)+T(n-1, k). Rows are the expansions of (1+2x)^k/(1-x)^(k+1).

G.f.: 1/(1-x-y-2xy). - Ralf Stephan, Apr 28 2004

T(n,k)=sum{j=0..n, C(k,j-k)*C(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}. [From Roger L. Bagula, Dec 09 2008]

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} comb(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

EXAMPLE

Rows begin

1 1 1 1 1 ...

1 4 7 10 13 ...

1 7 22 46 79 ...

1 10 46 136 307 ...

1 13 79 307 886 ...

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

Clear[a]; 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

CROSSREFS

Cf. 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.

Sequence in context: A131060 A124376 A047671 * A146986 A304141 A305047

Adjacent sequences:  A081574 A081575 A081576 * A081578 A081579 A081580

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Mar 23 2003

STATUS

approved

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Last modified July 15 04:34 EDT 2020. Contains 335763 sequences. (Running on oeis4.)