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.

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

Offset: 0

Author

Clark Kimberling, Aug 06 2011

Keywords

• nonn
• tabl

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. - Philippe Deléham, Oct 08 2011

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

Examples

			First six rows:
  1;
  1,  1;
  1,  3,  2;
  1,  4,  6,  3;
  1,  5, 10, 10,  4;
  1,  6, 15, 20, 15,  5;
		

Links

• 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.

Crossrefs

Cf. A193722, A153861, A193818.

Programs

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 *)

Formula

T(n,k) = A153861(n,n-k). - 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..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