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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
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

LINKS

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

EXAMPLE

First six rows:

2

3...5

4...7....9

5...9...12..14

6...11..15..18..20

7...13..18..22..25..27

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 *)

CROSSREFS

Cf. A193664, A193842, A193974.

Sequence in context: A099424 A117955 A074049 * A245057 A127521 A102399

Adjacent sequences:  A193970 A193971 A193972 * A193974 A193975 A193976

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 10 2011

STATUS

approved

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Last modified August 19 17:29 EDT 2017. Contains 290809 sequences.