A296180 - OEIS

%I #12 Dec 27 2017 10:05:27

%S 1,1,1,1,4,1,1,7,7,1,1,10,13,10,1,1,13,19,19,13,1,1,16,25,28,25,16,1,

%T 1,19,31,37,37,31,19,1,1,22,37,46,49,46,37,22,1,1,25,43,55,61,61,55,

%U 43,25,1,1,28,49,64,73,76,73,64,49,28,1

%N Triangle read by rows: T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n.

%C 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.

%C In general the recurrence is: T(m; n, 0) = 1 and T(m; n, n) = 1 for n >= 0; T(m; n, k) = (T(m; n-1, k-1)*T(m; n-1, k) + m)/T(m; n-2, k-1), for n >= 2, k = 1..n-1.

%C The general g.f. of the sequence of column k (with leading zeros) is G(m; k, x) = (x^k/(1 - x)^2)*(1 + (m*k - 1)*x), k >= 0.

%C 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}.

%C For a simple combinatorial interpretation see the one given in A130154 by _Rogério Serôdio_ which can be generalized to m >= 3.

%F T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n,

%F Recurrence: T(n, 0) = 1 and T(n, n) = 1 for n >= 0; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 3)/T(n-2, k-1), for n >= 2, k = 1..n-1.

%F G.f. of column k (with leading zeros): (x^k/(1 - x)^2)*(1 + (3*k-1)*x), k >= 0.

%F 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 = ...

%e The triangle T(n, k) begins:

%e n\k   0  1  2  3  4  5  6  7  8  9 10 ...

%e 0:    1

%e 1:    1  1

%e 2:    1  4  1

%e 3:    1  7  7  1

%e 4:    1 10 13 10  1

%e 5:    1 13 19 19 13  1

%e 6:    1 16 25 28 25 16  1

%e 7:    1 19 31 37 37 31 19  1

%e 8:    1 22 37 46 49 46 37 22  1

%e 9:    1 25 43 55 61 61 55 43 25  1

%e 10:   1 28 49 64 73 76 73 64 49 28  1

%e ...

%e Recurrence: 28 = T(6, 3) = (19*19 + 3)/13 = 28.

%t Table[3 k (n - k) + 1, {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 20 2017 *)

%o (PARI) lista(nn) = for(n=0, nn, for(k=0, n, print1(3*(n - k)*k + 1, ", "))) \\ _Iain Fox_, Dec 21 2017

%Y Cf. A077028, A130154.

%Y Columns (without leading zeros): A000012, A016777, A016921, A016921, A017173, A017533, ...

%K nonn,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Dec 20 2017