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A112351 Triangle read by rows, generated from (...5,3,1). 3
1, 1, 3, 1, 6, 5, 1, 9, 19, 7, 1, 12, 42, 44, 9, 1, 15, 74, 138, 85, 11, 1, 18, 115, 316, 363, 146, 13, 1, 21, 165, 605, 1059, 819, 231, 15, 1, 24, 224, 1032, 2470, 2984, 1652, 344, 17, 1, 27, 292, 1624, 4974, 8378, 7380, 3060 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A039755 (Analogs of a Stirling number of the second kind triangle); is generated through an analogous set of operations (but using the matrix M = [1 / 1 3 / 1 3 5 /...]). First few rows of the array are: 1, 3, 5, 7, 9, 11,... 1, 6, 19, 44, 85,... 1, 9, 42, 138, 363,... 1, 12, 74, 316, 1059,... ...

A112351 is jointly generated with A209414 as an array of coefficients of polynomials v(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica and Example sections. - Clark Kimberling, Mar 09 2012

Subtriangle of the triangle T(n,k) given by (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012

LINKS

Table of n, a(n) for n=0..52.

FORMULA

Let M = an infinite lower triangular matrix of the form [1 / 3 1 / 5 3 1 /...] (with the rest of the terms zeros). Perform M^n * [1 0 0 0...] forming an array. Antidiagonals of the array become rows of the triangle A112351.

From Philippe Deléham, Mar 12 2012: (Start)

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

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

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

EXAMPLE

The antidiagonal 1 9 19 7 of the array becomes row 3 of the triangle.

From Clark Kimberling, Mar 09 2012: (Start)

When jointly generated with A209414, the format as a triangle has the following first five rows:

1

1...3

1...6....5

1...9....19...7

1...12...42...44...9

1...15...74..138...85..11

The corresponding first five polynomials are

1

1 + 3x

1 + 6x + 5x^2

1 + 9x + 19x^2 + 7x^3

1 + 12x + 42x^2 + 44x^3 + 9x^4 (End)

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

1

1, 0

1, 3, 0

1, 6, 5, 0

1, 9, 19, 7, 0

1, 12, 42, 44, 9, 0

1, 15, 74, 138, 85, 11, 0

1, 18, 115, 316, 363, 146, 13, 0

- Philippe Deléham, Mar 12 2012

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_] := 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[%]    (* A209414 *)

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

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

TableForm[cv]

Flatten[%]    (* A112351 *)

(* Clark Kimberling, Mar 09 2012 *)

CROSSREFS

Cf. A039755, A005900 (array row 2), A061927 (array row 3), A209414.

Sequence in context: A116666 A208331 A061702 * A143858 A258993 A109954

Adjacent sequences:  A112348 A112349 A112350 * A112352 A112353 A112354

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 05 2005

STATUS

approved

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Last modified November 16 22:43 EST 2018. Contains 317275 sequences. (Running on oeis4.)