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A207628 Triangle of coefficients of polynomials v(n,x) jointly generated with A207627; see the Formula section. 4
1, 1, 4, 1, 6, 8, 1, 8, 20, 16, 1, 10, 36, 56, 32, 1, 12, 56, 128, 144, 64, 1, 14, 80, 240, 400, 352, 128, 1, 16, 108, 400, 880, 1152, 832, 256, 1, 18, 140, 616, 1680, 2912, 3136, 1920, 512, 1, 20, 176, 896, 2912, 6272, 8960, 8192, 4352, 1024, 1, 22, 216 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Column n is divisible by 2^(n-1); row n ends with 2^(n-1) for n > 2.

Also triangle T(n,k), k=0..n, read by rows, given by (1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 25 2012

Also seems to be square array of unsigned coefficients of 3U-2T (with T and U the two sequences of Chebyshev polynomials). - Thomas Baruchel, Jun 03 2018

LINKS

Table of n, a(n) for n=1..58.

FORMULA

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

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

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

The polynomials v(n,x) seem to be v(n)=sum(k=0,n-1, (-1)^(k+n+1) * x^k * polcoeff( 3*polchebyshev(2*n-k-2,2)-2*polchebyshev(2*n-k-2,1), k)) by using the PARI syntax. - Thomas Baruchel, Jun 05 2018

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

G.f.: (1+2*y*x)/(1-(1+2*y)*x). - Philippe Deléham, Feb 25 2012

T(n,k) = 2*T(n-1,k-1) + T(n-1,k) with T(0,0) = T(1,0) = 1, T(1,1) = 4. - Philippe Deléham, Feb 25 2012

As triangle T(n,k), k=0..n, it is given by T(n,k) = A029635(n,k)*2^k with T(0,0) = 1. - Philippe Deléham, Feb 25 2012

EXAMPLE

First five rows:

  1;

  1,  4;

  1,  6,  8;

  1,  8, 20, 16;

  1, 10, 36, 56, 32;

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] + 2 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[%]    (* A207627 *)

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

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

TableForm[cv]

Flatten[%]    (* A207628 *)

CROSSREFS

Cf. A207627.

Sequence in context: A066575 A070251 A134234 * A205137 A223534 A222391

Adjacent sequences:  A207625 A207626 A207627 * A207629 A207630 A207631

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 21 2012

STATUS

approved

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Last modified October 23 06:48 EDT 2019. Contains 328335 sequences. (Running on oeis4.)