A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

1, 1, 2, 2, 3, 4, 3, 8, 8, 8, 5, 15, 25, 20, 16, 8, 30, 55, 70, 48, 32, 13, 56, 125, 175, 184, 112, 64, 21, 104, 262, 440, 512, 464, 256, 128, 34, 189, 539, 1014, 1401, 1416, 1136, 576, 256, 55, 340, 1075, 2270, 3501, 4170, 3760, 2720, 1280, 512

Offset: 1

Author

Mark Dols, Sep 03 2009

Keywords

• nonn
• tabl

Comments

A164975 is jointly generated with A209125 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+ 2x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

Examples

			Triangle T(n,k), 0 <= k < n, n >= 1, begins:
   1;
   1,   2;
   2,   3,   4;
   3,   8,   8,   8;
   5,  15,  25,  20,  16;
   8,  30,  55,  70,  48,  32;
  13,  56, 125, 175, 184, 112,  64;
  21, 104, 262, 440, 512, 464, 256, 128;
  ...
T(7,1) = 30 + 2*8 + 15 - 5 = 56.
T(6,1) = 15 + 2*5 +  8 - 3 = 30.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A000045, A000079, A000244 (row sums).

Programs

Maple

A164975 := proc(n,k) option remember; if n <=0 or k > n or k< 1 then 0; elif k= 1 then combinat[fibonacci](n); else procname(n-1,k)+2*procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc: # R. J. Mathar, Jan 27 2011

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*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[%]    (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A164975 *)
(* Clark Kimberling, Mar 05 2012 *)
With[{nmax = 10}, Rest[CoefficientList[CoefficientList[Series[ x/(1 - 2*y*x-x-x^2+y*x^2), {x,0,nmax}, {y,0,nmax}], x], y]]//Flatten] (* G. C. Greubel, Jan 14 2018 *)

Formula

T(n,n-1) = A000079(n-1).

T(n,n-2) = A001792(n-2). - R. J. Mathar, Jan 27 2011

T(n,1) = A099920(n-1). - R. J. Mathar, Jan 27 2011

G.f.: x/(1-2*y*x-x-x^2+y*x^2). - Philippe Deléham, Mar 21 2012

Sum_{k=0..n-1, n>0} T(n,k)*x^k = A000045(n), A000244(n-1), A004254(n), A186446(n-1), A190980(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 21 2012

Extensions

Corrected by Philippe Deléham, Mar 21 2012