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A207611 Triangle of coefficients of polynomials v(n,x) jointly generated with A207610; see Formula section. 3
1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 8, 8, 5, 2, 1, 13, 15, 11, 6, 2, 1, 21, 28, 23, 14, 7, 2, 1, 34, 51, 47, 32, 17, 8, 2, 1, 55, 92, 93, 70, 42, 20, 9, 2, 1, 89, 164, 181, 148, 97, 53, 23, 10, 2, 1, 144, 290, 346, 306, 217, 128, 65, 26, 11, 2, 1, 233, 509, 653, 619, 472 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 1:  Fibonacci numbers, A000045

Column 2:  A029907

Row sums:  A003945.

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

Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

LINKS

Table of n, a(n) for n=1..71.

FORMULA

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

T(n,k) = T(n-1,k) + (n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n.

EXAMPLE

First five rows:

  1;

  2, 1;

  3, 2, 1;

  5, 4, 2, 1;

  8, 8, 5, 2, 1;

From Philippe Deléham, Mar 25 2012: (Start)

(0, 2, -1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, ...) begins:

  1;

  0,  1;

  0,  2,  1;

  0,  3,  2,  1;

  0,  5,  4,  2,  1;

  0,  8,  8,  5,  2,  1;

  0, 13, 15, 11,  6,  2,  1;

  0, 21, 28, 23, 14,  7,  2,  1; (End)

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*v[n - 1, x] + 1

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[%]    (* A207610 *)

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

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

TableForm[cv]

Flatten[%]    (* A207611 *)

PROG

(Python)

from sympy import Poly

from sympy.abc import x

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*v(n - 1, x) + 1

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

for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

CROSSREFS

Cf. A207610, A208510.

Sequence in context: A322083 A058399 A209434 * A320973 A058400 A131344

Adjacent sequences:  A207608 A207609 A207610 * A207612 A207613 A207614

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 19 2012

STATUS

approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)