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#12 by Peter Luschny at Wed Dec 27 10:05:27 EST 2017
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#11 by Michel Marcus at Thu Dec 21 04:49:19 EST 2017
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#10 by Michel Marcus at Thu Dec 21 04:49:10 EST 2017
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| COMMENTS
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This is member m = 3 of the family of triangles T(m; n, k) = m*(n - k)*k + 1, for m >= 0. For m = 0: : A000012(n, k) (read as a triangle); for m = 1: A077028 (rascal), for m = 2: T(2, n+1, k+1) = A130154(n, k). ). Motivated by A130154 to look at this family of triangles.
The general g.f. of the triangle T(m;, n, k) is GT(m; x, t) = (1 - (1 + t)*x + (m+1)*t*x^2)/((1 - t*x)*(1 - x))^2, and G(m; k, x) = (d/dt)^k GT(m; x, t)/k!|_{t=0}.
For a simple combinatorial interpretation see the one given in A130154 by _Rogério Serôdio_ which can be generalized to m >= 3.
A130154 by Rogério Serôdio which can be generalized to m >= 3.
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#9 by Wolfdieter Lang at Thu Dec 21 04:39:24 EST 2017
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#8 by Wolfdieter Lang at Thu Dec 21 04:38:04 EST 2017
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| FORMULA
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G.f. of triangle: (1 - (1 + t)*x + 4*t*x^2)/((1 - t*x)*(1 - x))^2 = 1 + (1+t)*x +(1 + 4*t + t^2)*x^2 + (1 + 7*t + 7*t^2 = . + t^3)*x^3 = ...
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Discussion
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| Thu Dec 21
| 04:39
| Wolfdieter Lang: To Michael De Vlieger: Thanks for your comment. Indeed, a wanted to give the first terms.
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#7 by Iain Fox at Thu Dec 21 00:27:54 EST 2017
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#6 by Iain Fox at Thu Dec 21 00:27:42 EST 2017
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| PROG
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(PARI) lista(nn) = for(n=0, nn, for(k=0, n, print1(3*(n - k)*k + 1, ", "))) \\ Iain Fox, Dec 21 2017
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#5 by Michael De Vlieger at Wed Dec 20 22:50:18 EST 2017
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#4 by Michael De Vlieger at Wed Dec 20 22:49:43 EST 2017
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Triangle read by rows: T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n,.
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| MATHEMATICA
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Table[3 k (n - k) + 1, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 20 2017 *)
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Discussion
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| Wed Dec 20
| 22:50
| Michael De Vlieger: Is last line of Formula missing something? (i.e., "G.f. of triangle: (1 - (1 + t)*x + 4*t*x^2)/((1 - t*x)*(1 - x))^2 = .")
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#3 by Wolfdieter Lang at Wed Dec 20 05:17:25 EST 2017
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