A193856 Triangular array: the fission of (p(n,x)) by ((2x+1)^n), where p(n,x)=(x+1)^n.

1, 1, 5, 1, 8, 19, 1, 11, 43, 65, 1, 14, 76, 194, 211, 1, 17, 118, 422, 793, 665, 1, 20, 169, 776, 2059, 3044, 2059, 1, 23, 229, 1283, 4387, 9221, 11191, 6305, 1, 26, 298, 1970, 8236, 22382, 38854, 39878, 19171, 1, 29, 376, 2864, 14146, 47090, 106000

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..51.

Formula

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = sum {i = 0..k} (-1)^k*binomial(n+1,k-i)*(-3)^(k-i) for 0 <= k <= n.

O.g.f.: 1/( (1 - 2*x*t)*(1 - (3*x + 1)*t) )= 1 + (1 + 5*x)*t + (1 + 8*x + 19*x^2)*t^2 + .... Cf. A193860.

The n-th row polynomial R(n,x) = 1/(x + 1)*( (3*x + 1)^(n+1) - (2*x)^(n+1) ). (End)

T(n, k) = 3^k*binomial(n+1, k)*hypergeom([1, -k], [n-k+2], 1/3). - Peter Luschny, Nov 19 2018

Example

First six rows:
1
1...5
1...8....19
1...11...43....65
1...14...76....194...211
1...17...118...422...793...665

Maple

T := (n, k) -> 3^k*binomial(n+1, k)*hypergeom([1, -k], [n-k+2], 1/3):
for n from 0 to 6 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Nov 19 2018

Mathematica

z = 10;
p[n_, x_] := (2 x + 1)^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}]]  (* A193856 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193857 *)

Crossrefs

Cf. A193842, A193857, A193860.

Sequence in context: A269229 A193089 A154310 * A255294 A115521 A140705

Adjacent sequences: A193853 A193854 A193855 * A193857 A193858 A193859

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 07 2011

Status

approved