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A207608 Triangle of coefficients of polynomials u(n,x) jointly generated with A207609; see the Formula section. 3
1, 2, 3, 3, 4, 11, 3, 5, 26, 20, 3, 6, 50, 74, 29, 3, 7, 85, 204, 149, 38, 3, 8, 133, 469, 547, 251, 47, 3, 9, 196, 952, 1618, 1160, 380, 56, 3, 10, 276, 1764, 4110, 4234, 2124, 536, 65, 3, 11, 375, 3048, 9318, 13036, 9262, 3520, 719, 74, 3, 12, 495, 4983 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 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..59.

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.

From Philippe Deléham, Mar 03 2012: (Start)

As triangle T(n,k), 0 <= k <= n:

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

G.f.: (1-y*x)/(1 - (2+y)*x - (y-1)*x^2).

Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A025192(n), A001077(n), A180038(n) for x = 0, 1, 2, 3 respectively. (End)

EXAMPLE

First five rows:

  1;

  2;

  3,  3;

  4, 11,  3;

  5, 26, 20,  3;

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

  1;

  2,   0;

  3,   3,   0;

  4,  11,   3,   0;

  5,  26,  20,   3,   0;

  6,  50,  74,  29,   3,   0;

  7,  85, 204, 149,  38,   3,   0;

  ... - Philippe Deléham, Mar 03 2012

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(u(n, x), x).all_coeffs()[::-1]

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

CROSSREFS

Cf. A207609.

Sequence in context: A173590 A128744 A293984 * A290818 A240376 A118963

Adjacent sequences:  A207605 A207606 A207607 * A207609 A207610 A207611

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Feb 19 2012

STATUS

approved

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Last modified October 22 18:02 EDT 2019. Contains 328319 sequences. (Running on oeis4.)