A193858 Triangular array: the fission of ((x+1)^n) by ((2x+1)^n).

1, 2, 3, 4, 10, 7, 8, 28, 34, 15, 16, 72, 124, 98, 31, 32, 176, 392, 444, 258, 63, 64, 416, 1136, 1672, 1404, 642, 127, 128, 960, 3104, 5616, 6152, 4092, 1538, 255, 256, 2176, 8128, 17440, 23536, 20488, 11260, 3586, 511, 512, 4864, 20608, 51136, 81952

Offset

0,2

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

From Andrew Howroyd, Feb 18 2024: (Start)

T(n,k) = Sum_{j=0..k} 2^(n-j) * binomial(n-j,k-j).

G.f.: A(x,y) = 1/(1 - (2 + 3*y)*x + 2*y*(1 + y)*x^2). (End)

Example

First six rows:
   1
   2     3
   4    10     7
   8    28    34    15
  16    72   124    98    31
  32   176   392   444   258   63

Maple

# The function 'fission' is defined in A193842.
A193858_row := n -> fission((n, x) -> (x+1)^n, (n, x) -> (2*x+1)^n, n);
for n from 0 to 5 do A193858_row(n) od; # Peter Luschny, Jul 23 2014

Mathematica

z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (2 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}]]  (* this sequence *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193859 *)

Prog

(Sage) # uses[fission from A193842]
A193858_row = lambda k: fission(lambda n, x: (x+1)^n, lambda n, x: (2*x+1)^n, k)
for n in range(7): A193858_row(n) # Peter Luschny, Jul 23 2014
(PARI) T(n, k)={sum(j=0, k, 2^(n-j) * binomial(n-j, k-j))} \\ Andrew Howroyd, Feb 18 2024

Crossrefs

Cf. A193842, A193859.

Sequence in context: A084190 A203070 A088303 * A054930 A329230 A119798

Adjacent sequences: A193855 A193856 A193857 * A193859 A193860 A193861

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 07 2011

Status

approved