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A210596 Triangle read by rows of coefficients of polynomials v(n,x) jointly generated with A210221; see the Formula section. 3
1, 1, 2, 2, 2, 4, 3, 6, 4, 8, 5, 10, 16, 8, 16, 8, 20, 28, 40, 16, 32, 13, 36, 64, 72, 96, 32, 64, 21, 66, 124, 184, 176, 224, 64, 128, 34, 118, 248, 376, 496, 416, 512, 128, 256, 55, 210, 476, 808, 1056, 1280, 960, 1152, 256, 512, 89, 370, 908, 1640, 2416 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n begins with F(n) and ends with 2^(n-1), where F = A000045 (Fibonacci numbers)

Row sums:  odd-indexed Fibonacci numbers, see A001519.

For a discussion and guide to related arrays, see A208510.

Riordan array (1/(1 - z - z^2), 2*z*(1 - z)/(1 - z - z^2)). - Peter Bala, Dec 30 2015

LINKS

G. C. Greubel, Rows n=1..102 of triangle, flattened

FORMULA

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

v(n,x) = u(n-1,x) + 2x*v(n-1,x),

where u(1,x) = 1, v(1,x) = 1.

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 25 2012

G.f.: 1/((1-x-x^2) - t*2*x*(1-x)). - G. C. Greubel, Dec 15 2018

EXAMPLE

First five rows:

  1

  1  2

  2  2  4

  3  6  4  8

  5 10 16  8 16

First three polynomials v(n,x): 1, 1 + 2x, 2 + 2x + 4x^2.

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] + 2 x*v[n - 1, x];

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

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

TableForm[cu]

Flatten[%]   (* A210221 *)

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

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

TableForm[cv]

Flatten[%]   (* A210596 *)

With[{m = 10}, CoefficientList[CoefficientList[Series[1/((1-x-x^2) - t*2*x*(1-x)), {x, 0, m}, {t, 0, m}], x], t]]//Flatten (* G. C. Greubel, Dec 15 2018 *)

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) + 2*x*v(n - 1, x)

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

for n in xrange(1, 13): print a(n)

(PARI) {T(n, k) = if(n==1 && k==0, 1, if(n==2 && k==0, 1, if(n==2 && k==1, 2, if(k<0 || k>n-1, 0, T(n-1, k) + 2*T(n-1, k-1) + T(n-2, k) - 2*T(n-2, k-1) ))))};

for(n=1, 15, for(k=0, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 15 2018

CROSSREFS

Cf. A000045, A000079, A001519, A210221, A208510.

Sequence in context: A029145 A238999 A097986 * A240078 A228660 A228796

Adjacent sequences:  A210593 A210594 A210595 * A210597 A210598 A210599

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 24 2012

STATUS

approved

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Last modified November 12 13:43 EST 2019. Contains 329058 sequences. (Running on oeis4.)