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A144903
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)).
11
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
OFFSET
0,13
LINKS
FORMULA
G.f. of column k: x/((1-x-x^3)*(1-x)^(k-1)).
A(n, n) = A144904(n).
From G. C. Greubel, Aug 01 2022: (Start)
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.
T(2*n, n) = A144904(n). (End)
EXAMPLE
Square array (A(n,k)) begins:
0, 0, 0, 0, 0, 0, 0 ... A000004;
1, 1, 1, 1, 1, 1, 1 ... A000012;
0, 1, 2, 3, 4, 5, 6 ... A001477;
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;
MAPLE
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);
MATHEMATICA
(* First program *)
a[n_, k_] := SeriesCoefficient[x/((1-x-x^3)*(1-x)^(k-1)), {x, 0, n}];
Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}]//Flatten (* Jean-François Alcover, Jan 15 2014 *)
(* 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 *)
PROG
(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]]) >;
[A144903(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 01 2022
(SageMath)
def A000930(n): return sum(binomial(n-2*j, j) for j in (0..(n//3)))
def A144903(n, k):
if (k==0): return 0
else: return sum(binomial(n-k+j-2, j)*A000930(k-j-1) for j in (0..k-1))
flatten([[A144903(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Aug 01 2022
CROSSREFS
Rows 0-4, 6 give: A000004, A000012, A001477, A000217, A050407(n+3), A027658.
Columns 0-9 give: A078012 and A135851(n+2), A078012(n+2) and A135851(n+4), A077868(n-1) for n>0, A050228(n-1) for n>0, A226405, A144898, A144899, A144900, A144901, A144902.
Main diagonal gives: A144904.
Cf. A000930.
Sequence in context: A060959 A342689 A077042 * A356266 A108934 A108947
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 24 2008
STATUS
approved