|
%I #54 Jan 25 2024 11:50:38
%S 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,
%T 1,19,121,307,307,121,19,1,1,22,172,586,886,586,172,22,1,1,25,232,
%U 1000,2086,2086,1000,232,25,1,1,28,301,1576,4258,5944,4258,1576,301,28,1
%N Pascal-(1,2,1) array read by antidiagonals.
%C 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.
%C 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
%C All entries are == 1 mod 3. - _Roger L. Bagula_, Oct 04 2008
%C Row sums are A002605. - _Roger L. Bagula_, Dec 09 2008)
%C As a number triangle T, T(2n,n)=A069835(n). - _Philippe Deléham_, Jan 10 2014
%H Reinhard Zumkeller, <a href="/A081577/b081577.txt">Rows n = 0..125 of table, flattened</a>
%H Peter Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry1/barry411.html">The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
%H Shanghua Zheng, Li Guo, and Huizhen Qiu, <a href="https://arxiv.org/abs/2401.11363">Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras</a>, arXiv:2401.11363 [math.RA], 2024. See pp. 44-45.
%F 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).
%F Rows are the expansions of (1+2*x)^k/(1-x)^(k+1).
%F G.f.: 1/(1-x-y-2*x*y). - _Ralf Stephan_, Apr 28 2004
%F T(n,k) = Sum_{j=0..n} binomial(k,j-k)*binomial(n+k-j,k)*2^(j-k). - _Paul Barry_, Oct 23 2006
%F a(n) = 2*{0, a(n-2), 0} + {0, a(n-1)} + {a(n-1), 0}. - _Roger L. Bagula_, Dec 09 2008
%F T(n, k) = Hypergeometric2F1([-k, k-n], [1], 3). - _Jean-François Alcover_, May 24 2013
%F 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
%F Sum_{k=0..n} T(n,k) = A002605(n). - _G. C. Greubel_, May 25 2021
%e Square array begins as:
%e 1, 1, 1, 1, 1, ... A000012;
%e 1, 4, 7, 10, 13, ... A016777;
%e 1, 7, 22, 46, 79, ... A038764;
%e 1, 10, 46, 136, 307, ... A081583;
%e 1, 13, 79, 307, 886, ... A081584;
%e From _Roger L. Bagula_, Dec 09 2008: (Start)
%e As a triangle this begins:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 7, 7, 1;
%e 1, 10, 22, 10, 1;
%e 1, 13, 46, 46, 13, 1;
%e 1, 16, 79, 136, 79, 16, 1;
%e 1, 19, 121, 307, 307, 121, 19, 1;
%e 1, 22, 172, 586, 886, 586, 172, 22, 1;
%e 1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1;
%e 1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1; (End)
%t 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 *)
%t Table[Hypergeometric2F1[-k, k-n, 1, 3], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, May 24 2013 *)
%o (Haskell)
%o a081577 n k = a081577_tabl !! n !! k
%o a081577_row n = a081577_tabl !! n
%o a081577_tabl = map fst $ iterate
%o (\(us, vs) -> (vs, zipWith (+) (map (* 2) ([0] ++ us ++ [0])) $
%o zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
%o -- _Reinhard Zumkeller_, Mar 16 2014
%o (Magma)
%o A081577:= func< n,k | (&+[Binomial(k,j)*Binomial(n-j,k)*2^j: j in [0..n-k]]) >;
%o [A081577(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 25 2021
%o (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
%Y Cf. A002605, A081578, A081579, A081580.
%Y 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.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Mar 23 2003
|
