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A264774
Triangle T(n,k) = binomial(5*n - 4*k, 4*n - 3*k), 0 <= k <= n.
3
1, 5, 1, 45, 6, 1, 455, 55, 7, 1, 4845, 560, 66, 8, 1, 53130, 5985, 680, 78, 9, 1, 593775, 65780, 7315, 816, 91, 10, 1, 6724520, 736281, 80730, 8855, 969, 105, 11, 1, 76904685, 8347680, 906192, 98280, 10626, 1140, 120, 12, 1, 886163135, 95548245, 10295472, 1107568, 118755, 12650, 1330, 136, 13, 1
OFFSET
0,2
COMMENTS
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.
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.
LINKS
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Section 2, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.
FORMULA
T(n,k) = binomial(5*n - 4*k, n - k).
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.
EXAMPLE
Triangle begins
n\k | 0 1 2 3 4 5 6 7
------+---------------------------------------------
0 | 1
1 | 5 1
2 | 45 6 1
3 | 455 55 7 1
4 | 4845 560 66 8 1
5 | 53130 5985 680 78 9 1
6 | 593775 65780 7315 816 91 10 1
7 | 6724520 736281 80730 8855 969 105 11 1
...
MAPLE
A264774:= proc(n, k) binomial(5*n - 4*k, 4*n - 3*k); end proc:
seq(seq(A264774(n, k), k = 0..n), n = 0..10);
MATHEMATICA
Table[Binomial[5 n - 4 k, 4 n - 3 k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)
PROG
(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
CROSSREFS
Cf. A001449 (column 0), A079589(column 1). Cf. A002294, A007318, A092392 (C(2n-k,n)), A113139, A119301 (C(3n-k,n-k)), A264772, A264773.
Sequence in context: A221366 A134274 A134275 * A114154 A297899 A134273
KEYWORD
nonn,tabl,easy,changed
AUTHOR
Peter Bala, Nov 30 2015
STATUS
approved