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A264774 Triangle T(n,k) = binomial(5*n - 4*k, 4*n - 3*k), 0 <= k <= n.

%I A264774 #31 Nov 16 2024 18:47:58
%S A264774 1,5,1,45,6,1,455,55,7,1,4845,560,66,8,1,53130,5985,680,78,9,1,593775,
%T A264774 65780,7315,816,91,10,1,6724520,736281,80730,8855,969,105,11,1,
%U A264774 76904685,8347680,906192,98280,10626,1140,120,12,1,886163135,95548245,10295472,1107568,118755,12650,1330,136,13,1
%N A264774 Triangle T(n,k) = binomial(5*n - 4*k, 4*n - 3*k), 0 <= k <= n.
%C A264774 Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + ... is the o.g.f. for A002294 and f(x) = g(x)/(5 - 4*g(x)) = 1 + 5*x + 45*x^2 + 455*x^3 + 4845*x^4 + ... is the o.g.f. for A001449.
%C A264774 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 = 4 and b = 3. See A092392, A264772, A264773 and A113139 for further examples.
%H A264774 Peter Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>
%H A264774 E. Lebensztayn, <a href="https://doi.org/10.46298/dmtcs.512">On the asymptotic enumeration of accessible automata, Section 2</a>, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
%H A264774 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 A264774 T(n,k) = binomial(5*n - 4*k, n - k).
%F A264774 O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(5*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(4*n + 1)*binomial(5*n,n)*x^n.
%e A264774 Triangle begins
%e A264774   n\k |       0      1     2    3   4   5   6   7
%e A264774 ------+---------------------------------------------
%e A264774    0  |       1
%e A264774    1  |       5      1
%e A264774    2  |      45      6     1
%e A264774    3  |     455     55     7    1
%e A264774    4  |    4845    560    66    8   1
%e A264774    5  |   53130   5985   680   78   9   1
%e A264774    6  |  593775  65780  7315  816  91  10   1
%e A264774    7  | 6724520 736281 80730 8855 969 105  11  1
%e A264774 ...
%p A264774 A264774:= proc(n,k) binomial(5*n - 4*k, 4*n - 3*k); end proc:
%p A264774 seq(seq(A264774(n,k), k = 0..n), n = 0..10);
%t A264774 Table[Binomial[5 n - 4 k, 4 n - 3 k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 01 2015 *)
%o A264774 (Magma) /* As triangle */ [[Binomial(5*n-4*k, 4*n-3*k): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Dec 02 2015
%Y A264774 Cf. A001449 (column 0), A079589(column 1). Cf. A002294, A007318, A092392 (C(2n-k,n)), A113139, A119301 (C(3n-k,n-k)), A264772, A264773.
%K A264774 nonn,tabl,easy
%O A264774 0,2
%A A264774 _Peter Bala_, Nov 30 2015