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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 (list; table; graph; refs; listen; history; text; internal format)
 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 R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, Section 5.6 CreateSpace Independent Publishing Platform 2006, 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 Adjacent sequences:  A264771 A264772 A264773 * A264775 A264776 A264777 KEYWORD nonn,tabl,easy AUTHOR Peter Bala, Nov 30 2015 STATUS approved

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Last modified April 3 18:40 EDT 2020. Contains 333198 sequences. (Running on oeis4.)