%I #30 Feb 06 2024 05:15:29

%S 1,3,1,15,4,1,84,21,5,1,495,120,28,6,1,3003,715,165,36,7,1,18564,4368,

%T 1001,220,45,8,1,116280,27132,6188,1365,286,55,9,1,735471,170544,

%U 38760,8568,1820,364,66,10,1,4686825,1081575,245157,54264,11628,2380,455,78,11,1

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

%C Riordan array (f(x), x*g(x)), where g(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... is the o.g.f. for A001764 and f(x) = g(x)/(3 - 2*g(x)) = 1 + 3*x + 15*x^2 + 84*x^3 + 495*x^4 + ... is the o.g.f. for A005809.

%C The even numbered columns give the Riordan array A119301, the odd numbered columns give the Riordan array A144484. A159841 is the array formed from columns 1,4,7,10,....

%C More generally, if R = (R(n,k))n,k>=0 is a proper Riordan array, 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 = 2, b = 1. See A092392, A264773, A264774 and A113139 for further examples.

%H Michael De Vlieger, <a href="/A264772/b264772.txt">Table of n, a(n) for n = 0..11475</a>

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

%H Paul Barry, <a href="https://arxiv.org/abs/1906.06373">On the halves of a Riordan array and their antecedents</a>, arXiv:1906.06373 [math.CO], 2019.

%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(3*n - 2*k, n - k).

%F O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(3*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(2*n + 1)*binomial(3*n,n)*x^n.

%e Triangle begins

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

%e ----------------------------------------------

%e ..0..| 1

%e ..1..| 3 1

%e ..2..| 15 4 1

%e ..3..| 84 21 5 1

%e ..4..| 495 120 28 6 1

%e ..5..| 3003 715 165 36 7 1

%e ..6..| 18564 4368 1001 220 45 8 1

%e ..7..| 116280 27132 6188 1365 286 55 9 1

%e ...

%p A264772:= proc(n,k) binomial(3*n - 2*k, 2*n - k); end proc:

%p seq(seq(A264772(n,k), k = 0..n), n = 0..10);

%t Table[Binomial[3 n - 2 k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 01 2015 *)

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

%Y Cf. A005809 (column 0), A045721 (column 1), A025174 (column 2), A004319 (column 3), A236194 (column 4), A013698 (column 5). Cf. A001764, A007318, A092392, A119301 (C(3n-k,2n)), A144484 (C(3n+1-k,2n+1)), A159841 (C(3n+1,2n+k+1)), A264773, A264774.

%K nonn,easy,tabl

%O 0,2

%A _Peter Bala_, Nov 24 2015