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A209419 Triangle of coefficients of polynomials u(n,x) jointly generated with A209420; see the Formula section. 3
1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 5, 17, 21, 10, 1, 8, 35, 58, 45, 15, 1, 13, 68, 144, 154, 85, 21, 1, 21, 129, 330, 452, 350, 147, 28, 1, 34, 239, 719, 1198, 1195, 714, 238, 36, 1, 55, 436, 1506, 2959, 3611, 2799, 1344, 366, 45, 1, 89, 785, 3063, 6930, 10005, 9537, 5985, 2376, 540, 55, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Column 1:  Fibonacci numbers (A000045)

Alternating row sums: (1,0,0,0,0,0,0,0,0,0,0,0,...)

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

Triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

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

v(n,x) = (x+1)*u(n-1,x) + (x+1)*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) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 26 2012

G.f.: x*(1 - x*y)/(1 - x - x^2 - 2*y*x + y^2*x^2). - G. C. Greubel, Jan 03 2018

EXAMPLE

First five rows:

  1;

  1,  1;

  2,  3,  1;

  3,  8,  6,  1;

  5, 17, 21, 10,  1;

First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A209419 *)

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

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

TableForm[cv]

Flatten[%]    (* A209420 *)

CoefficientList[CoefficientList[Series[(1*x - x^2*y)/(1 - x - x^2 - 2*y*x + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

CROSSREFS

Cf. A209420, A208510.

Sequence in context: A214978 A295380 A093768 * A119011 A300866 A130477

Adjacent sequences:  A209416 A209417 A209418 * A209420 A209421 A209422

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 19 12:22 EDT 2019. Contains 328221 sequences. (Running on oeis4.)