A233508 Numerators of the triangle of polynomial coefficients P(0,x)=1, 2*P(n)=(1+x)*((1+x)^(n-1)+x^(n-1)). Of the first array of A133135.

1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 3, 5, 1, 1, 5, 5, 5, 3, 1, 1, 3, 15, 10, 15, 7, 1, 1, 7, 21, 35, 35, 21, 4, 1, 1, 4, 14, 28, 35, 28, 14, 9, 1, 1, 9, 18, 42, 63, 63, 42, 18, 5, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 11, 1

Offset: 0

Author

Paul Curtz, Dec 11 2013

Keywords

• nonn
• tabl
• frac

Comments

Discovered via Euler polynomials A060096(n)/A060097(n).

The fractional sequence is 1, 1, 1, 1/2, 3/2, 1, 1/2, 3/2, 2, 1, 1/2, 2, 3, 5/2, 1,... =a(n)/b(n). There is a correspondant sequence for Bernoulli polynomials (*).

Examples

			1,
1, 1,
1, 3, 1,
1, 3, 2, 1,
1, 2, 3, 5, 1,
1, 5, 5, 5, 3, 1, etc.
		

Crossrefs

Cf. (*) A193815.

Programs

Mathematica

p[n_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t[n_, k_] := Coefficient[p[n], x, k] // Numerator; Table[t[n, k], {n, 0, 10 }, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

Formula

a(n) = reduced A133138(n)/A007395.