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A207609 Triangle of coefficients of polynomials v(n,x) jointly generated with A207608; see Formula section. 3
1, 1, 3, 1, 8, 3, 1, 15, 17, 3, 1, 24, 54, 26, 3, 1, 35, 130, 120, 35, 3, 1, 48, 265, 398, 213, 44, 3, 1, 63, 483, 1071, 909, 333, 53, 3, 1, 80, 812, 2492, 3074, 1744, 480, 62, 3, 1, 99, 1284, 5208, 8802, 7138, 2984, 654, 71, 3, 1, 120, 1935, 10020, 22230, 24408, 14370, 4710, 855, 80, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Subtriangle of the triangle given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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

v(n,x)=2x*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(n-2,k), n>2. - Philippe Deléham, Mar 03 2012

Sum_{k, 0<=k<=n, n>=1} T(n,k)*x^k = A000012(n), A052156(n-1), A048876(n-1) for x = 0, 1, 2 respectively. - Philippe Deléham, Mar 03 2012

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

EXAMPLE

First five rows:

1

1...3

1...8....3

1...15...17...3

1...24...54...26...3

Triangle (1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -2, 0, 0, 0, ...) begins :

1

1, 0

1, 3, 0

1, 8, 3, 0

1, 15, 17, 3, 0

1, 24, 54, 26, 3, 0

1, 35, 130, 120, 35, 3, 0

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_] := 2 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[%]    (* A207608 *)

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

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

TableForm[cv]

Flatten[%]    (* A207609 *)

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 2*x*u(n - 1, x) + (x + 1)*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) # Indranil Ghosh, May 28 2017

CROSSREFS

Cf. A207608.

Sequence in context: A179449 A049541 A249757 * A130300 A065451 A178148

Adjacent sequences:  A207606 A207607 A207608 * A207610 A207611 A207612

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 19 2012

STATUS

approved

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Last modified October 17 10:26 EDT 2018. Contains 316276 sequences. (Running on oeis4.)