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.

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, 33, 35, 9, 17, 24, 30, 35, 39, 42, 44, 10, 19, 27, 34, 40, 45, 49, 52, 54, 11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 12, 23, 33, 42, 50, 57, 63, 68, 72, 75, 77, 13, 25, 36, 46

Offset: 0

Author

Clark Kimberling, Aug 10 2011

Keywords

• nonn
• tabl
• easy

Comments

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

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

Examples

			First six rows:
  2;
  3,  5;
  4,  7,  9;
  5,  9, 12, 14;
  6, 11, 15, 18, 20;
  7, 13, 18, 22, 25, 27;
  ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000217, A193664, A193842, A193974.

Programs

Maple

a000217 := proc(n) n*(n+1)/2 end:
seq(print(seq(a000217(n+2) - a000217(n+1-k),k=0..n)),n=0..5); # Georg Fischer, May 03 2022

Mathematica

z = 13;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193973 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193974 *)

Formula

T(n, k) = A000217(n + 2) - A000217(n + 1 - k), 0 <= k <= n. - Georg Fischer, May 03 2022