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A207605 Triangle of coefficients of polynomials u(n,x) jointly generated with A106195; see the Formula section. 3
1, 2, 4, 1, 8, 4, 1, 16, 12, 5, 1, 32, 32, 18, 6, 1, 64, 80, 56, 25, 7, 1, 128, 192, 160, 88, 33, 8, 1, 256, 448, 432, 280, 129, 42, 9, 1, 512, 1024, 1120, 832, 450, 180, 52, 10, 1, 1024, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 2048, 5120, 6912, 6400, 4424, 2364, 985, 316, 75, 12, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums: 1,2,5,13,... (odd-indexed Fibonacci numbers).

Alternating row sums: 1,2,3,5,... (Fibonacci numbers).

Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

LINKS

Table of n, a(n) for n=1..67.

FORMULA

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1.

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(1,0) = 1, T(2,0) = 2, T(2,1) = 0. - Philippe Deléham, Mar 22 2012

G.f.: -(-1+x*y)*x*y/(1-x*y-2*x+x^2*y). - R. J. Mathar, Aug 11 2015

T(n,k) = [x^k] Sum_{k=0..n} binomial(n, k)*hypergeom([-k, n-k], [-n], x). - Peter Luschny, Feb 16 2018

EXAMPLE

First five rows:

   1

   2

   4   1

   8   4   1

  16  12   5   1

  32  32  18   6   1

First four polynomials u(n,x): 1, 2, 4 + x, 8 + 4x + x^2.

(1, 1, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:

   1

   1,  0

   2,  0,  0

   4,  1,  0,  0

   8,  4,  1,  0,  0

  16, 12,  5,  1,  0,  0

  32, 32, 18,  6,  1,  0,  0. - Philippe Deléham, Mar 22 2012

MAPLE

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):

T := (n, k) -> binomial(n, k)*hypergeom([-k, n-k], [-n], x):

P := [seq(add(simplify(T(n, k)), k=0..n), n=0..11)]:

seq(CoeffList(p), p in P); # Peter Luschny, Feb 16 2018

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + v[n - 1, x]

v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]

Table[Factor[u[n, x]], {n, 1, z}]

Table[Factor[v[n, x]], {n, 1, z}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]  (* A207605 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]  (* A106195 *)

PROG

(Python)

from sympy import Poly

def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)

def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)

def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]

for n in xrange(1, 13): print a(n) # Indranil Ghosh, May 27 2017

CROSSREFS

Cf. A106195.

Sequence in context: A232723 A275486 A065278 * A112931 A121685 A125810

Adjacent sequences:  A207602 A207603 A207604 * A207606 A207607 A207608

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Feb 19 2012

STATUS

approved

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Last modified October 16 03:34 EDT 2019. Contains 328040 sequences. (Running on oeis4.)