A146769 Coefficients of polynomial P(n) by rows, with P(n) = (x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1) for n > 0 and P(0) = 1.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 12, 18, 12, 1, 1, 25, 50, 50, 25, 1, 1, 54, 135, 180, 135, 54, 1, 1, 119, 357, 595, 595, 357, 119, 1, 1, 264, 924, 1848, 2310, 1848, 924, 264, 1, 1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1, 1, 1290, 5805, 15480, 27090, 32508

Offset: 0

Author

Roger L. Bagula, Nov 02 2008

Keywords

• nonn
• tabl

Comments

Original name: A new symmetrical polynomial form to give a triangle sequence: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].

Row sums are:{1, 2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840}.

Row sums are 1 and (6*2^k + 4^k)/8 for k >= 1 (see A257273). - Robert Israel, Apr 29 2015

Examples

			1;
1,    1;
1,    3,    1;
1,    6,    6,     1;
1,   12,   18,    12,     1;
1,   25,   50,    50,    25,     1;
1,   54,  135,   180,   135,    54,     1;
1,  119,  357,   595,   595,   357,   119,     1;
1,  264,  924,  1848,  2310,  1848,   924,   264,    1;
1,  585, 2340,  5460,  8190,  8190,  5460,  2340,  585,    1;
1, 1290, 5805, 15480, 27090, 32508, 27090, 15480, 5805, 1290, 1;
...
		

Crossrefs

Cf. A257273 (row sums).

Programs

Magma

/* As triangle: */ [1]; for n in [1..10] do; R := PolynomialAlgebra(RationalField(), n); Coefficients((x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1)); end for; // Bruno Berselli, Apr 30 2015

Mathematica

p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

Formula

G.f.: y/(4*(2*y-1)) - 1/(x*y+y-1) - 1/(8*(2*x*y+2*y-1)) + 1/(8*(2*x*y-1)). - Robert Israel, Apr 29 2015

Extensions

New name from Charles R Greathouse IV, Apr 29 2015