%I #29 Feb 06 2024 10:18:30

%S 1,4,1,28,5,1,220,36,6,1,1820,286,45,7,1,15504,2380,364,55,8,1,134596,

%T 20349,3060,455,66,9,1,1184040,177100,26334,3876,560,78,10,1,10518300,

%U 1560780,230230,33649,4845,680,91,11,1,94143280,13884156,2035800,296010,42504,5985,816,105,12,1

%N Triangle T(n,k) = binomial(4*n - 3*k, 3*n - 2*k), 0 <= k <= n.

%C Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + ... is the o.g.f. for A002293 and f(x) = g(x)/(4 - 3*g(x)) = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + ... is the o.g.f. for A005810.

%C More generally, if (R(n,k))n,k>=0 is a proper Riordan array and m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 3 and b = 2. See A092392, A264772, A264774 and A113139 for further examples.

%H Paolo Xausa, <a href="/A264773/b264773.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).

%H Peter Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>

%H E. Lebensztayn, <a href="https://doi.org/10.46298/dmtcs.512">On the asymptotic enumeration of accessible automata</a>, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.

%H R. Sprugnoli, <a href="https://web.archive.org/web/20170401103408/http://www.dsi.unifi.it/~resp/Handbook.pdf">An Introduction to Mathematical Methods in Combinatorics</a> CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

%F T(n,k) = binomial(4*n - 3*k, n - k).

%F O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(4*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(3*n + 1)*binomial(4*n,n)*x^n.

%e Triangle begins

%e n\k | 0 1 2 3 4 5 6 7

%e ------+-----------------------------------------------

%e 0 | 1

%e 1 | 4 1

%e 2 | 28 5 1

%e 3 | 220 36 6 1

%e 4 | 1820 286 45 7 1

%e 5 | 15504 2380 364 55 8 1

%e 6 | 134596 20349 3060 455 66 9 1

%e 7 | 1184040 177100 26334 3876 560 78 10 1

%e ...

%p A264773:= proc(n,k) binomial(4*n - 3*k, 3*n - 2*k); end proc:

%p seq(seq(A264773(n,k), k = 0..n), n = 0..10);

%t A264773[n_,k_] := Binomial[4*n - 3*k, n - k];

%t Table[A264773[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Feb 06 2024 *)

%o (Magma) /* As triangle: */ [[Binomial(4*n-3*k, 3*n-2*k): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Dec 02 2015

%Y A005810 (column 0), A052203 (column 1), A257633 (column 2), A224274 (column 3), A004331 (column 4). Cf. A002293, A007318, A092392 (C(2n-k,n)), A119301 (C(3n-k,n-k)), A264772, A264774.

%K nonn,tabl,easy

%O 0,2

%A _Peter Bala_, Nov 30 2015