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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. 3
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
FORMULA
T(n, k) = A000217(n + 2) - A000217(n + 1 - k), 0 <= k <= n. - Georg Fischer, May 03 2022
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
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 *)
CROSSREFS
Sequence in context: A117955 A074049 A326777 * A245057 A127521 A350583
KEYWORD
nonn,tabl,changed
AUTHOR
Clark Kimberling, Aug 10 2011
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)