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A081577
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Pascal-(1,2,1) array read by antidiagonals.
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30
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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).
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
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
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EXAMPLE
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Square array begins as:
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)
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MATHEMATICA
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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 *)
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PROG
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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])
(Magma)
A081577:= func< n, k | (&+[Binomial(k, j)*Binomial(n-j, k)*2^j: j in [0..n-k]]) >;
(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
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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