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%I #21 Jan 29 2019 04:35:35
%S 0,0,1,0,1,1,0,1,3,1,0,1,5,9,1,0,1,7,21,29,1,0,1,9,37,85,99,1,0,1,11,
%T 57,177,341,351,1,0,1,13,81,313,807,1365,1275,1,0,1,15,109,501,1593,
%U 3579,5461,4707,1,0,1,17,141,749,2811,7737,15591,21845,17577,1
%N A(n, k) = [x^k] (1 - 4*x)^(-n/2)*x/(1 - x), square array read by ascending antidiagonals with n >= 0 and k >= 0.
%C General asymptotic formula for g.f. (1 - 4*x)^(-j/2)*x/(1 - x) and fixed j>0 is a(n) ~ n^(j/2 - 1) * 4^n / (3*Gamma(j/2)). - _Vaclav Kotesovec_, Jan 29 2019
%F For n>0 and k>0 let X(n, k) denote the set of all tuples of length n with elements from {0, ..., k-1} with sum < k. Let b(m) = binomial(2*m, m). Then A(n, k) = Sum_{(j1,...,jn) in X(n, k)} b(j1)*b(j2)*...*b(jn).
%e [n\k] 0 1 2 3 4 5 6 7 8 9
%e -------------------------------------------------------------------
%e [0] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A057427
%e [1] 0, 1, 3, 9, 29, 99, 351, 1275, 4707, 17577, ... A006134
%e [2] 0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, ... A002450
%e [3] 0, 1, 7, 37, 177, 807, 3579, 15591, 67071, 285861, ... A277178
%e [4] 0, 1, 9, 57, 313, 1593, 7737, 36409, 167481, 757305, ... A014916
%e [5] 0, 1, 11, 81, 501, 2811, 14823, 74883, 366603, 1752273, ... A323223
%e [6] 0, 1, 13, 109, 749, 4589, 26093, 140781, 730605, 3679725, ...
%e [7] 0, 1, 15, 141, 1065, 7071, 43107, 247311, 1355847, 7175661, ...
%e [8] 0, 1, 17, 177, 1457, 10417, 67761, 411825, 2377905, 13191345, ...
%e [9] 0, 1, 19, 217, 1933, 14803, 102319, 656587, 3982195, 23104441, ...
%e Triangle given by antidiagonals:
%e 0;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 3, 1;
%e 0, 1, 5, 9, 1;
%e 0, 1, 7, 21, 29, 1;
%e 0, 1, 9, 37, 85, 99, 1;
%e 0, 1, 11, 57, 177, 341, 351, 1;
%e 0, 1, 13, 81, 313, 807, 1365, 1275, 1;
%e 0, 1, 15, 109, 501, 1593, 3579, 5461, 4707, 1;
%p Row := proc(n, len) local ogf, ser; ogf := (1 - 4*x)^(-n/2)*x/(1 - x);
%p ser := series(ogf, x, (n+1)*len+1); seq(coeff(ser, x, j), j=0..len) end:
%p for n from 0 to 9 do Row(n, 9) od;
%t BF[N_, K_] := Module[{}, r[n_, k_] := FrobeniusSolve[ConstantArray[1, n], k];
%t X[n_] := Flatten[Table[r[N, j], {j, 0, n - 1}], 1];
%t CentralBinomial[n_] := Binomial[2 n, n];
%t Sum[Product[CentralBinomial[m[[i]]], {i, 1, N}], {m , X[K]}]];
%t Trow[n_] := Table[BF[n, k], {k, 0, 9}]; Table[Trow[n], {n, 1, 9}]
%Y Sums of antidiagonals are A323217. Main diagonal is A323219.
%Y Rows: A057427 (n=0), A006134 (n=1), A002450 (n=2), A277178 (n=3), A014916 (n=4), A323223 (n=5).
%Y Columns: A005408 (k=2), A059993 (k=3), A323218 (k=4).
%Y Similar array based on Catalan numbers is A323224.
%K nonn,tabl
%O 0,9
%A _Peter Luschny_, Jan 24 2019
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