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A144903
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of x/((1-x-x^3)*(1-x)^(k-1)).
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11
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0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 3, 2, 1, 0, 1, 4, 6, 5, 3, 1, 0, 1, 5, 10, 11, 8, 4, 2, 0, 1, 6, 15, 21, 19, 12, 6, 3, 0, 1, 7, 21, 36, 40, 31, 18, 9, 4, 0, 1, 8, 28, 57, 76, 71, 49, 27, 13, 6, 0, 1, 9, 36, 85, 133, 147, 120, 76, 40, 19, 9, 0, 1, 10, 45, 121, 218, 280, 267, 196, 116, 59, 28, 13
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OFFSET
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0,13
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LINKS
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FORMULA
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G.f. of column k: x/((1-x-x^3)*(1-x)^(k-1)).
A(n, k) = Sum_{j=0..n-1} binomial(k+j-2, j)*A000930(n-j-1), with A(0, k) = 0.
T(n, k) = Sum_{j=0..k-1} binomial(n-k-j-2, j)*A000930(k-j-1), with T(n, 0) = 0.
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EXAMPLE
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Square array (A(n,k)) begins:
0, 1, 3, 6, 10, 15, 21 ... A000217;
1, 2, 5, 11, 21, 36, 57 ... A050407;
1, 3, 8, 19, 40, 76, 133 ... ;
1, 4, 12, 31, 71, 147, 200 ... A027658;
Antidiagonal triangle (T(n,k)) begins as:
0;
0, 1;
0, 1, 0;
0, 1, 1, 0;
0, 1, 2, 1, 1;
0, 1, 3, 3, 2, 1;
0, 1, 4, 6, 5, 3, 1;
0, 1, 5, 10, 11, 8, 4, 2;
0, 1, 6, 15, 21, 19, 12, 6, 3;
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MAPLE
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A:= proc(n, k) coeftayl (x/ (1-x-x^3)/ (1-x)^(k-1), x=0, n) end:
seq(seq(A(n, d-n), n=0..d), d=0..13);
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MATHEMATICA
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(* First program *)
a[n_, k_] := SeriesCoefficient[x/((1-x-x^3)*(1-x)^(k-1)), {x, 0, n}];
(* Second Program *)
A000930[n_]:= A000930[n]= Sum[Binomial[n-2*j, j], {j, 0, Floor[n/3]}];
T[n_, k_]:= T[n, k]= If[k==0, 0, Sum[Binomial[n-k+j-2, j]*A000930[k-j-1], {j, 0, k- 1}]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 01 2022 *)
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PROG
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(Magma)
A000930:= func< n | (&+[Binomial(n-2*j, j): j in [0..Floor(n/3)]]) >;
A144903:= func< n, k | k eq 0 select 0 else (&+[Binomial(n-k+j-2, j)*A000930(k-j-1) : j in [0..k-1]]) >;
(SageMath)
def A000930(n): return sum(binomial(n-2*j, j) for j in (0..(n//3)))
if (k==0): return 0
else: return sum(binomial(n-k+j-2, j)*A000930(k-j-1) for j in (0..k-1))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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