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This a weighted generalized Catalan sequence triangle (A365673) with the hexagonal numbers as weights.

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This a weighted generalized Catalan sequence (A365673) with the hexagonal numbers as weights.

Cf. A000384, A126156 (main diagonal), A365673 (general case).

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proposed

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[1] 1, 1;

[2] 1, 7, 7;

[3] 1, 22, 139, 139;

[4] 1, 50, 889, 5473, 5473;

[5] 1, 95, 3549, 58708, 357721, 357721;

[6] 1, 161, 10794, 360940, 5771821, 34988647, 34988647;

allocated for Peter LuschnyTriangle read by rows. T(n, k) = 1 if k = 0, equals T(n, k-1) if k = n, and otherwise is (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k).

1, 1, 1, 1, 7, 7, 1, 22, 139, 139, 1, 50, 889, 5473, 5473, 1, 95, 3549, 58708, 357721, 357721, 1, 161, 10794, 360940, 5771821, 34988647, 34988647, 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619

0,5

This triangle is described by Peter Bala (see link).

Peter Bala, <a href="/A126156/a126156.pdf">A triangle for calculating A126156</a>.

Triangle T(n, k) starts:

[0] 1;

[1] 1, 1;

[2] 1, 7, 7;

[3] 1, 22, 139, 139;

[4] 1, 50, 889, 5473, 5473;

[5] 1, 95, 3549, 58708, 357721, 357721;

[6] 1, 161, 10794, 360940, 5771821, 34988647, 34988647;

[7] 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619;

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k) fi fi end:

Cf. A126156 (main diagonal).

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nonn,tabl

Peter Luschny, Sep 29 2023

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allocated for Peter Luschny

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