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A209420 Triangle of coefficients of polynomials v(n,x) jointly generated with A209419; see the Formula section. 3
1, 2, 2, 3, 6, 3, 5, 14, 13, 4, 8, 30, 41, 24, 5, 13, 60, 109, 96, 40, 6, 21, 116, 262, 308, 196, 62, 7, 34, 218, 590, 868, 743, 364, 91, 8, 55, 402, 1267, 2240, 2413, 1604, 630, 128, 9, 89, 730, 2627, 5424, 7046, 5926, 3186, 1032, 174, 10, 144, 1310, 5299, 12516, 19040, 19382, 13255, 5928, 1617, 230 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 1:  Fibonacci numbers (A000045).

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

Triangle given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 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) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 26 2012

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

EXAMPLE

First five rows:

  1;

  2,  2;

  3,  6,  3;

  5, 14, 13,  4;

  8, 30, 41, 24,  5;

First three polynomials v(n,x): 1, 2 + 2x, 3 + 6x + 3x^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[x*(1 + x)/(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. A209419, A208510.

Sequence in context: A308503 A196967 A210859 * A317449 A222310 A294033

Adjacent sequences:  A209417 A209418 A209419 * A209421 A209422 A209423

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 23 05:56 EDT 2019. Contains 328335 sequences. (Running on oeis4.)