A146899 An additive term polynomial as a stand-alone polynomial: t(n,m) = binomial(n, m)/2 if binomial(n, m) is even, binomial(n, m) + 1 otherwise; p(x,n) = (Sum_{m=1..n-1} t(n, m)*x^m*(1 + x^(n - 2*m)))/(2*x).

1, 4, 4, 2, 3, 2, 6, 5, 5, 6, 3, 16, 10, 16, 3, 8, 22, 36, 36, 22, 8, 4, 14, 28, 35, 28, 14, 4, 10, 18, 42, 63, 63, 42, 18, 10, 5, 46, 60, 105, 126, 105, 60, 46, 5, 12, 56, 166, 165, 231, 231, 165, 166, 56, 12, 6, 33, 110, 496, 396, 462, 396, 496, 110, 33, 6

Offset

2,2

Comments

Row sums begin 1, 8, 7, 22, 48, 132, 127, 266, 558, ....

Links

Table of n, a(n) for n=2..67.

Formula

t(n,m) = binomial(n, m)/2 if binomial(n, m) is even, binomial(n, m) + 1 otherwise;

p(x,n) = (Sum_{m=1..n-1} t(n, m)*x^m*(1 + x^(n - 2*m)))/(2*x);

t(n,m) = coefficients(p(x,n)).

Example

Table begins
   1;
   4,  4;
   2,  3,  2;
   6,  5,  5,   6;
   3, 16, 10,  16,   3;
   8, 22, 36,  36,  22,   8;
   4, 14, 28,  35,  28,  14,  4;
  10, 18, 42,  63,  63,  42, 18, 10;
   5, 46, 60, 105, 126, 105, 60, 46, 5;

Mathematica

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

Prog

(PARI) t(n, m) = my(x=binomial(n, m)); if (x%2, x+1, x/2);
p(n) = sum(m=1, n-1, t(n, m)*x^m*(1 + x^(n - 2*m)))/(2*x);
row(n) = Vec(p(n)); \\ Michel Marcus, Jan 27 2021

Crossrefs

Cf. A007318 (binomial).

Sequence in context: A161778 A099655 A276149 * A317726 A031351 A068923

Adjacent sequences: A146896 A146897 A146898 * A146900 A146901 A146902

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 02 2008

Extensions

More terms from Michel Marcus, Jan 27 2021

Status

approved