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A208331 Triangle of coefficients of polynomials v(n,x) jointly generated with A208330; see the Formula section. 3
1, 1, 3, 1, 6, 5, 1, 9, 15, 11, 1, 12, 30, 44, 21, 1, 15, 50, 110, 105, 43, 1, 18, 75, 220, 315, 258, 85, 1, 21, 105, 385, 735, 903, 595, 171, 1, 24, 140, 616, 1470, 2408, 2380, 1368, 341, 1, 27, 180, 924, 2646, 5418, 7140, 6156, 3069, 683, 1, 30, 225 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

LINKS

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

FORMULA

u(n,x)=u(n-1,x)+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) = A001045(n+2)*binomial(n-1,k). - Philippe Deléham, Mar 18 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012

EXAMPLE

First five rows:

1

1...3

1...6...5

1...9...15...11

1...12...30...44...21

First five polynomials u(n,x):

1, 1 + 3x, 1 + 6x + 5x^2, 1 + 9x + 15x^2 + 11x^3, 1+12x + 30x^2 + 44x^3 + 21x^4.

(1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, ...) begins :

1

1, 0

1, 3, 0

1, 6, 5, 0

1, 9, 15, 11, 0

1, 12, 30, 44, 21, 0. - Philippe Deléham, Mar 18 2012

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

v[n_, x_] := 2 x*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[%]  (* A208330 *)

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

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

TableForm[cv]

Flatten[%]  (* A208331 *)

CROSSREFS

Cf. A208330.

Sequence in context: A308948 A225246 A116666 * A061702 A112351 A143858

Adjacent sequences:  A208328 A208329 A208330 * A208332 A208333 A208334

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)