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A153861 Triangle read by rows, binomial transform of triangle A153860. 5
1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 10, 10, 5, 1, 5, 15, 20, 15, 6, 1, 6, 21, 35, 35, 21, 7, 1, 7, 28, 56, 70, 56, 28, 8, 1, 8, 36, 84, 126, 126, 84, 36, 9, 1, 9, 45, 120, 210, 252, 210, 120, 45, 10, 1, 10, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums = A095121: (1, 2, 6, 14, 30, 62, 126,...).

Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009

A123110*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 06 2009

A153861 is the fusion of polynomial sequences p(n,x)=x^n+x^(n-1)+...+x+1 and q(n,x)=(x+1)^n; see A193722 for the definition of fusion. - Clark Kimberling, Aug 06 2011

LINKS

G. C. Greubel, Table of n, a(n) for the first 46 rows

FORMULA

Triangle read by rows, A007318 * A153860. Remove left two columns of Pascal's triangle and append (1, 1, 2, 3, 4, 5,...).

As a recursive operation by way of example, row 3 = (3, 6, 4, 1) =

[1, 1, 1, 0] * (flipped Pascal's triangle matrix) = [1, 3, 3, 1]

[1, 2, 1, 0]

[1, 1, 0, 0]

[1, 0, 0, 0].

(Cf. analogous operation in A130405, but in A153861 the linear multiplier = [1,1,1,...,0].)

T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)-T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0)=2, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 15 2013

G.f.: (1-x+x^2+x^2*y)/((x-1)*(-1+x+x*y)). - R. J. Mathar, Aug 11 2015

EXAMPLE

First few rows of the triangle are:

1;

1, 1;

2, 3, 1;

3, 6, 4, 1;

4, 10, 10, 5, 1;

5, 15, 20, 15, 6, 1;

6, 21, 35, 35, 21, 7, 1;

7, 28, 56, 70, 56, 28, 8, 1;

8, 36, 84, 126, 126, 84, 36, 9, 1;

9, 45, 120, 210, 252, 210, 120, 45, 10, 1;

...

MATHEMATICA

z = 10; c = 1; d = 1;

p[0, x_] := 1

p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;

q[n_, x_] := (c*x + d)^n

t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

g[n_] := CoefficientList[w[n, x], {x}]

TableForm[Table[Reverse[g[n]], {n, -1, z}]]

Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193815 *)

TableForm[Table[g[n], {n, -1, z}]]

Flatten[Table[g[n], {n, -1, z}]]   (* A153861 *)

(* Clark Kimberling, Aug 06 2011 *)

CROSSREFS

Cf. A153860, A095121, A130405.

Sequence in context: A086404 A192852 A152976 * A118981 A117938 A256193

Adjacent sequences:  A153858 A153859 A153860 * A153862 A153863 A153864

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jan 03 2009

STATUS

approved

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Last modified July 19 12:35 EDT 2019. Contains 325159 sequences. (Running on oeis4.)