%I #21 Mar 18 2024 17:05:45
%S 2,3,5,4,7,9,5,9,12,14,6,11,15,18,20,7,13,18,22,25,27,8,15,21,26,30,
%T 33,35,9,17,24,30,35,39,42,44,10,19,27,34,40,45,49,52,54,11,21,30,38,
%U 45,51,56,60,63,65,12,23,33,42,50,57,63,68,72,75,77,13,25,36,46
%N Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=x*p(n-1,x)+1 with p(0,x)=1.
%C See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%C This array show the differences of the sequence of triangular numbers A000217); viz., row n consists of t(n) - t(n-k) for k=1..n-1. - _Clark Kimberling_, Apr 15 2017
%H Clark Kimberling, <a href="/A193973/b193973.txt">Table of n, a(n) for n = 0..10000</a>
%F T(n, k) = A000217(n + 2) - A000217(n + 1 - k), 0 <= k <= n. - _Georg Fischer_, May 03 2022
%e First six rows:
%e 2
%e 3...5
%e 4...7....9
%e 5...9...12..14
%e 6...11..15..18..20
%e 7...13..18..22..25..27
%p a000217 := proc(n) n*(n+1)/2 end:
%p seq(print(seq(a000217(n+2) - a000217(n+1-k),k=0..n)),n=0..5); # _Georg Fischer_, May 03 2022
%t z = 13;
%t p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
%t q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
%t p1[n_, k_] := Coefficient[p[n, x], x^k];
%t p1[n_, 0] := p[n, x] /. x -> 0;
%t d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t h[n_] := CoefficientList[d[n, x], {x}]
%t TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A193973 *)
%t TableForm[Table[h[n], {n, 0, z}]]
%t Flatten[Table[h[n], {n, -1, z}]] (* A193974 *)
%Y Cf. A000217, A193664, A193842, A193974.
%K nonn,tabl,changed
%O 0,1
%A _Clark Kimberling_, Aug 10 2011
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