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
Examples
First six rows: 1 2 3 4 10 7 8 28 34 15 16 72 124 98 31 32 176 392 444 258 63
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Programs
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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
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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 *) -
PARI
T(n,k)={sum(j=0, k, 2^(n-j) * binomial(n-j, k-j))} \\ Andrew Howroyd, Feb 18 2024 -
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
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)
Comments