A193846 Triangular array: the fission of ((x+2)^n) by ((x+1)^n).

2, 4, 8, 8, 28, 26, 16, 80, 136, 80, 32, 208, 512, 568, 242, 64, 512, 1648, 2672, 2188, 728, 128, 1216, 4832, 10288, 12392, 8020, 2186, 256, 2816, 13312, 35072, 55648, 53216, 28432, 6560, 512, 6400, 35072, 110080, 216512, 273376, 216512, 98416

Offset

0,1

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

G.f.: A(x,y) = 2/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2). - Andrew Howroyd, Feb 18 2024

Example

First six rows:
2
4....8
8....28....26
16...80....136....80
32...208...512....568....242
64...512...1648...2672...2188...728

Mathematica

p[n_, x_] := (x + 2)^n;
q[n_, x_] := (x + 1)^n
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}]]   (* A193846 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193847 *)
TableForm[Table[Reverse[h[n]/2], {n, 0, z}]]
Flatten[Table[Reverse[h[n]]/2, {n, -1, z}]] (* A193848 *)
TableForm[Table[h[n]/2, {n, 0, z}]]
Flatten[Table[h[n]/2, {n, -1, z}]]  (* A193849 *)

Prog

(PARI) T(n)={[2*Vecrev(p) | p<-Vec(1/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 18 2024

Crossrefs

Cf. A193842, A193847, A193848.

Sequence in context: A140119 A273068 A362532 * A255908 A341107 A344075

Adjacent sequences: A193843 A193844 A193845 * A193847 A193848 A193849

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 07 2011

Status

approved