A264773 -id:A264773 - cp's OEIS frontend

A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.

1, 3, 1, 15, 4, 1, 84, 21, 5, 1, 495, 120, 28, 6, 1, 3003, 715, 165, 36, 7, 1, 18564, 4368, 1001, 220, 45, 8, 1, 116280, 27132, 6188, 1365, 286, 55, 9, 1, 735471, 170544, 38760, 8568, 1820, 364, 66, 10, 1, 4686825, 1081575, 245157, 54264, 11628, 2380, 455, 78, 11, 1

Offset: 0

Peter Bala, Nov 24 2015

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.
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,....
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.

			Triangle begins
.n\k.|......0.....1....2....3...4..5...6..7...
----------------------------------------------
..0..|      1
..1..|      3     1
..2..|     15     4    1
..3..|     84    21    5    1
..4..|    495   120   28    6   1
..5..|   3003   715  165   36   7  1
..6..|  18564  4368 1001  220  45  8  1
..7..| 116280 27132 6188 1365 286 55  9  1
...

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Peter Bala, A 4-parameter family of embedded Riordan arrays
Paul Barry, On the halves of a Riordan array and their antecedents, arXiv:1906.06373 [math.CO], 2019.
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.

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.

/* As triangle */ [[Binomial(3*n-2*k, n-k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264772:= proc(n,k) binomial(3*n - 2*k, 2*n - k); end proc:
seq(seq(A264772(n,k), k = 0..n), n = 0..10);

Table[Binomial[3 n - 2 k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

T(n,k) = binomial(3*n - 2*k, n - k).

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.

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

Original entry on oeis.org

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

Peter Bala, Nov 30 2015

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.

			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
...

Peter Bala, A 4-parameter family of embedded Riordan arrays
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.

Cf. A001449 (column 0), A079589(column 1). Cf. A002294, A007318, A092392 (C(2n-k,n)), A113139, A119301 (C(3n-k,n-k)), A264772, A264773.

/* As triangle */ [[Binomial(5*n-4*k, 4*n-3*k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

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);

Table[Binomial[5 n - 4 k, 4 n - 3 k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

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.

cp's OEIS Frontend

A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.

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

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cp's OEIS Frontend

A264772 Triangle T(n,k) = binomial(3*n - 2*k, 2*n - k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.

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