cp's OEIS Frontend

A193973 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.

%I A193973 #24 Feb 15 2025 20:26:40
%S A193973 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 A193973 33,35,9,17,24,30,35,39,42,44,10,19,27,34,40,45,49,52,54,11,21,30,38,
%U A193973 45,51,56,60,63,65,12,23,33,42,50,57,63,68,72,75,77,13,25,36,46
%N A193973 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 A193973 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%C A193973 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 A193973 Clark Kimberling, <a href="/A193973/b193973.txt">Table of n, a(n) for n = 0..10000</a>
%F A193973 T(n, k) = A000217(n + 2) - A000217(n + 1 - k), 0 <= k <= n. - _Georg Fischer_, May 03 2022
%e A193973 First six rows:
%e A193973   2;
%e A193973   3,  5;
%e A193973   4,  7,  9;
%e A193973   5,  9, 12, 14;
%e A193973   6, 11, 15, 18, 20;
%e A193973   7, 13, 18, 22, 25, 27;
%e A193973   ...
%p A193973 a000217 := proc(n) n*(n+1)/2 end:
%p A193973 seq(print(seq(a000217(n+2) - a000217(n+1-k),k=0..n)),n=0..5); # _Georg Fischer_, May 03 2022
%t A193973 z = 13;
%t A193973 p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
%t A193973 q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
%t A193973 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193973 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193973 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193973 h[n_] := CoefficientList[d[n, x], {x}]
%t A193973 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193973 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193973 *)
%t A193973 TableForm[Table[h[n], {n, 0, z}]]
%t A193973 Flatten[Table[h[n], {n, -1, z}]]  (* A193974 *)
%Y A193973 Cf. A000217, A193664, A193842, A193974.
%K A193973 nonn,tabl,easy
%O A193973 0,1
%A A193973 _Clark Kimberling_, Aug 10 2011