A264773 Triangle T(n,k) = binomial(4*n - 3*k, 3*n - 2*k), 0 <= k <= n.

1, 4, 1, 28, 5, 1, 220, 36, 6, 1, 1820, 286, 45, 7, 1, 15504, 2380, 364, 55, 8, 1, 134596, 20349, 3060, 455, 66, 9, 1, 1184040, 177100, 26334, 3876, 560, 78, 10, 1, 10518300, 1560780, 230230, 33649, 4845, 680, 91, 11, 1, 94143280, 13884156, 2035800, 296010, 42504, 5985, 816, 105, 12, 1

Offset: 0

Author

Peter Bala, Nov 30 2015

Keywords

• nonn
• tabl
• easy

Comments

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.

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.

Examples

			Triangle begins
  n\k |       0      1     2    3   4   5   6   7
------+-----------------------------------------------
   0  |       1
   1  |       4      1
   2  |      28      5     1
   3  |     220     36     6    1
   4  |    1820    286    45    7   1
   5  |   15504   2380   364   55   8   1
   6  |  134596  20349  3060  455  66   9   1
   7  | 1184040 177100 26334 3876 560  78  10   1
...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
• Peter Bala, A 4-parameter family of embedded Riordan arrays
• E. Lebensztayn, On the asymptotic enumeration of accessible automata, 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.

Crossrefs

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.

Programs

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

Maple

A264773:= proc(n,k) binomial(4*n - 3*k, 3*n - 2*k); end proc:
seq(seq(A264773(n,k), k = 0..n), n = 0..10);

Mathematica

A264773[n_,k_] := Binomial[4*n - 3*k, n - k];
Table[A264773[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 06 2024 *)

Formula

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

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.