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%I #26 Feb 15 2022 02:08:39
%S 1,1,2,2,5,3,3,10,11,4,4,17,27,19,5,5,26,54,56,29,6,6,37,95,130,100,
%T 41,7,7,50,153,260,265,162,55,8,8,65,231,469,595,483,245,71,9,9,82,
%U 332,784,1190,1204,812,352,89,10
%N A128132 * A007318.
%C A007318 * A128132 = A128133. Row sums = A128135: (1, 3, 10, 28, 72, 176, ...).
%F A128132 * A007318 as infinite lower triangular matrices (assuming the top of the Pascal triangle A007318 is shifted from (0,0) to (1,1)).
%F From _Petros Hadjicostas_, Jul 26 2020: (Start)
%F T(n,k) = n*binomial(n-1, k-1) - binomial(n-2, k-1)*[n <> k] for 1 <= k <= n, where [ ] is the Iverson bracket.
%F Bivariate o.g.f.: x*y*(1 - x + x^2*(1 + y))/(1 - x*(1 + y))^2.
%F T(n,k) = T(n-1,k) + T(n-1,k-1) + binomial(n-1,k-1) for 2 <= k <= n with (n,k) <> (2,2).
%F T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-2,k-2) + 2*T(n-1,k-1) - 2*T(n-2,k-1) for 3 <= k <= n with (n,k) <> (3,3).
%F T(n,1) = n - 1 for n >= 2.
%F T(n,2) = A002522(n-1) for n >= 2.
%F T(n,3) = A164845(n-3) for n >= 3.
%F T(n,4) = A332697(n-3) for n >= 4.
%F T(n,n) = n for n >= 1.
%F T(n,n-1) = A028387(n-2) for n >= 2. (End)
%e Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
%e 1;
%e 1, 2;
%e 2, 5, 3;
%e 3, 10, 11, 4;
%e 4, 17, 27, 19, 5;
%e 5, 26, 54, 56, 29, 6;
%e 6, 37, 95, 130, 100, 41, 7;
%e ...
%Y Cf. A002522, A007318, A028387, A128132, A128133, A128135, A164845, A332697.
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Feb 15 2007
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