OFFSET
0,2
COMMENTS
Riordan array (1/(1-x)^2, 2*x/(1-x)). - Philippe Deléham, Jan 29 2014
Let P(n, x) := Sum_{k=1..n} T(n, k)*x^k. Then P(n, P(m, x)) = P(n*m, x) for all n and m in Z. - Michael Somos, Apr 10 2020
FORMULA
T(n, k) = binomial(n, k)/2 * 2^k. - Michael Somos, Apr 10 2020
EXAMPLE
First six rows:
1
2...2
3...6....4
4...12...16...8
5...20...40...40....16
6...30...80...120...96...32
Production matrix begins
2......2
-1/2...1...2
1/4....0...1...2
-1/8...0...0...1...2
1/16...0...0...0...1...2
-1/32..0...0...0...0...1...2
1/64...0...0...0...0...0...1...2
-1/128.0...0...0...0...0...0...1...2
1/256..0...0...0...0...0...0...0...1...2
- Philippe Deléham, Jan 29 2014
MATHEMATICA
z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
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}]] (* A115068 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]] (* A193862 *)
T[ n_, k_] := Binomial[n, k]/2 2^k; (* Michael Somos, Apr 10 2020 *)
PROG
(PARI) {T(n, k) = binomial(n, k)/2 * 2^k}; /* Michael Somos, Apr 10 2020 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 07 2011
STATUS
approved