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A264773
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Triangle T(n,k) = binomial(4*n - 3*k, 3*n - 2*k), 0 <= k <= n.
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3
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
...
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MAPLE
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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);
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MATHEMATICA
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A264773[n_, k_] := Binomial[4*n - 3*k, n - k];
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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