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A193815 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=x^n+x^(n-1)+...+x+1 and q(n,x)=(x+1)^n. 8
1, 1, 1, 1, 3, 2, 1, 4, 6, 3, 1, 5, 10, 10, 4, 1, 6, 15, 20, 15, 5, 1, 7, 21, 35, 35, 21, 6, 1, 8, 28, 56, 70, 56, 28, 7, 1, 9, 36, 84, 126, 126, 84, 36, 8, 1, 10, 45, 120, 210, 252, 210, 120, 45, 9, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 10, 1, 12, 66, 220, 495 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (1,0,-1,1,0,0,0,0,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - From Philippe Deléham, Oct 08 2011.

Row sums are A095121 .- Philippe Deléham, Nov 24 2011

LINKS

Table of n, a(n) for n=0..70.

Branko Malesevic, Yue Hu, Cristinel Mortici, Accurate Estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit, arXiv:1801.04963 [math.CA], 2018.

FORMULA

T(n,k)=A153861(n,n-k). - From Philippe Deléham, Oct 08 2011.

G.f.: (1-y*x+y*(y+1)*x^2)/((1-y*x)*(1-(y+1)*x)). - Philippe Deléham, Nov 24 2011

Sum_{k, 0<=k<=n} T(n,k)*x^k = (x+1)*((x+1)^n - x^n)+0^n. - Philippe Deléham, Nov 24 2011

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

EXAMPLE

First six rows:

1

1....1

1....3....2

1....4....6....3

1....5....10...10...4

1....6....15...20...15...5

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 *)

t[0, 0] = t[1, 0] = t[1, 1] = t[2, 0] = 1; t[2, 1] = 3; t[2, 2] = 2; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = t[n-1, k]+2*t[n-1, k-1]-t[n-2, k-1]-t[n-2, k-2]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013, after Philippe Deléham *)

CROSSREFS

Cf. A193722, A153861, A193818.

Sequence in context: A210797 A222220 A271830 * A104509 A271513 A306801

Adjacent sequences:  A193812 A193813 A193814 * A193816 A193817 A193818

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 06 2011

STATUS

approved

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Last modified July 21 18:13 EDT 2019. Contains 325199 sequences. (Running on oeis4.)