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A358628
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Square array A(i,j), i >= 0, j >= 0, read by antidiagonals: A(i,j) = Sum_{|X|=0..i} Sum_{|Y|=0..i} Product_{k=1..j} (1+X(k)+Y(k)), where X and Y are multi-indices of length j.
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1
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1, 1, 1, 1, 8, 1, 1, 23, 27, 1, 1, 46, 176, 64, 1, 1, 77, 640, 800, 125, 1, 1, 116, 1707, 4850, 2675, 216, 1, 1, 163, 3761, 19607, 25235, 7301, 343, 1, 1, 218, 7282, 61216, 147952, 101528, 17248, 512, 1, 1, 281, 12846, 159854, 635376, 831600, 338688, 36576, 729, 1
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OFFSET
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0,5
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COMMENTS
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Here X = (X(1),...,X(j)) and |X| = Sum_{k=1..j} X(j).
For 0 <= n <= i, let f_{n}(t) = Sum_{k=0..n} c(n,k)*t^k denote a polynomial of degree n containing at most n+1 terms. Let T_1, ..., T_j be independent variables and define B(i,j) as the set of all multivariate polynomials of the form Prod_{k=1..j} f_{a_k}(T_k), where a = (a_1,...,a_j) denotes a j-tuple of nonnegative integers satisfying Sum_{k=1..j} a_k <= i. Set N = |B(i,j)| and let (g_n)_{n=1..N} denote an ordering of the elements of B(i,j). Let S(m,n) denote the (maximum) number of terms in the expansion of g_m*g_n. Then A(i,j) = Sum_{m=1..N} Sum_{n=1..N} S(m,n).
A(i,j) <= (i+1)^(3*j) for all i >= 0, j >= 0.
It is conjectured that A(i,j) = binomial(i+j,j)^2 * p_j(i) for some polynomial function p_j(i) of degree j.
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LINKS
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FORMULA
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G.f. for column j (conjectured): Sum_{k=0..2*j} binomial(2*j,k)^2 * x^k / (1-x)^(3*j+1)
G.f.: Integral_{t=0..Pi} (1-2*sqrt(x)*cos(t)+x)/((1-2*sqrt(x)*cos(t)+x)^2 - y(1-x)) dt / Pi.
B(x,y,0):=1, B(x,y,z):=Sum_{a=0..x} Sum_{b=0..y} (1+a+b)*B(x-a,y-b,z-1); A(i,j) = B(i,i,j).
A(n,1) = binomial(n+1,1)^2 * (n+1) = A000578(n+1).
A(n,2) = binomial(n+2,2)^2 * (7*n^2 + 21*n + 18)/18.
A(n,3) = binomial(n+3,3)^2 * (n+2)*(11*n^2 + 44*n + 60)/120.
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EXAMPLE
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The square array A(i,j) (i >= 0, j >= 0) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 8, 23, 46, 77, 116, 163, ...
1, 27, 176, 640, 1707, 3761, 7282, ...
1, 64, 800, 4850, 19607, 61216, 159854, ...
1, 125, 2675, 25235, 147952, 635376, 2191100, ...
1, 216, 7301, 101528, 831600, 4783008, 21359228, ...
1, 343, 17248, 338688, 3755808, 28261728, 160542734, ...
...
Example computation of A(1,2): let f_0(t) = 1 and f_1(t) = 1+t. We have B(1,2) = {1,1+T_1,1+T_2} and N = 3. The table below gives the (maximum) number of terms in the expansions of pairwise-products of elements of B(1,2):
1 1+T_1 1+T_2
1 1 2 2
1+T_1 2 3 4
1+T_2 2 4 3
The sum of these entries is A(1,2) = 23.
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MATHEMATICA
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A[i_, j_] :=(If[j>0, mults={}; Do[lst=FrobeniusSolve[ConstantArray[1, j], k]; Do[AppendTo[mults, lst[[m]]], {m, 1, Length[lst]}], {k, 0, i}]; a=0; Do[Do[a=a+Times @@ (1+x+y), {y, mults}], {x, mults}]; a, 1]); Flatten[Table[A[i, j-i], {j, 0, 9}, {i, 0, j}]]
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PROG
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(MATLAB)
function A = EvaluateA(i, j)
% Construct all multi-indices of length j and total degree <= i.
v = zeros(1, j);
N = nchoosek(i+j, j);
W = zeros(N, j);
count = 1;
while ~isempty(v)
v(j) = v(j)+1;
for k = j:(-1):2
if v(k) > i
v(k) = 0;
v(k-1) = v(k-1)+1;
end
end
if sum(v) <= i
count = count+1;
W(count, 1:j) = v;
elseif v(1) > i
break
end
end
% Allocate outgoing value of A and loop over all pairs of multi-indices.
A = 0;
for count_X = 1:N
vX = W(count_X, 1:j);
for count_Y = 1:N
vY = W(count_Y, 1:j);
A = A + prod(1+vX+vY);
end
end
end
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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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