A163762 Triangle of coefficients of polynomials H(n,x)=(U^n+L^n)/2+(U^n-L^n)/(2d), where U=x+d, L=x-d, d=(x+4)^(1/2).

1, 1, 1, 1, 3, 4, 1, 6, 13, 4, 1, 10, 29, 24, 16, 1, 15, 55, 81, 88, 16, 1, 21, 95, 207, 300, 144, 64, 1, 28, 154, 448, 813, 684, 496, 64, 1, 36, 238, 868, 1913, 2352, 2272, 768, 256, 1, 45, 354, 1554, 4077, 6625, 7984, 4704, 2560, 256, 1, 55, 510, 2622, 8061, 16283

Offset

1,5

Comments

H(n,x)=P(n,x)+Q(n,x), where P and Q are given by A162516, A162517.

H(n,0)=4^Floor(n/2) for n=0,1,2,...

H(n,1)=A063727(n); row sums

(Column 2)=A000217 (triangular numbers)

Links

Table of n, a(n) for n=1..61.

Formula

H(n,x)=2*x*H(n-1,x)-(x^2-x-4)*H(n-2,x), where H(0,x)=1, H(1,x)=x+1.

H(n,x)=(1+1/d)*U^n+(1-1/d)*L^n, where U=x+d, L=x-d, d=(x+4)^(1/2).

Example

First six rows:
1
1...1
1...3...4
1...6..13...4
1..10..29..24..16
1..15..55..81..88..16
Row 6 represents x^5+15*x^4+55*x^3+81*x^2+88*x+16.

Crossrefs

A000217, A063727, A162516, A162517.

Sequence in context: A348733 A368471 A351569 * A347084 A226776 A339278

Adjacent sequences: A163759 A163760 A163761 * A163763 A163764 A163765

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2009

Status

approved