A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.

1, 1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 13, 20, 16, 1, 6, 19, 38, 48, 32, 1, 7, 26, 63, 104, 112, 64, 1, 8, 34, 96, 192, 272, 256, 128, 1, 9, 43, 138, 321, 552, 688, 576, 256, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512, 1, 11, 64, 253, 743, 1683, 2972, 4048

Offset

0,3

Comments

Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section.

Row sums: (1,3,8,...), even-indexed Fibonacci numbers.

Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers.

v(n,2) = A107839(n), v(n,n) = 2^(n-1), v(n+1,n) = A001792(n),

v(n+2,n) = A049611, v(n+3,n) = A049612.

Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012

Essentially triangle in A049600. - Philippe Deléham, Mar 23 2012

Links

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened

Formula

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1.

As DELTA-triangle with 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 12 2012

G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - Philippe Deléham, Mar 12 2012

T(n,k) = A106195(n-1,n-k), k = 1..n. - Reinhard Zumkeller, Dec 16 2013

From Peter Bala, Aug 11 2015: (Start)

The following remarks assume the row and column indexing start at 0.

T(n,k) = Sum_{i = 0..k} 2^(k-i)*binomial(n-k,i)*binomial(k,i) = Sum_{i = 0..k} binomial(n-k+i,i)*binomial(k,i).

Riordan array (1/(1 - x), x*(2 - x)/(1 - x)).

O.g.f. 1/(1 - (2*t + 1)*x + t*x^2) = 1 + (1 + 2*t)*x + (1 + 3*t + 4*t^2)*x^2 + ....

Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End)

For k<n, T(n,k) = T(n-1,k) + Sum_{i=1..k} T(n-i,k-i). - Glen Whitney, Aug 17 2021

Example

First five rows:
  1;
  1, 2;
  1, 3,  4;
  1, 4,  8,  8;
  1, 5, 13, 20, 16;
First five polynomials v(n,x):
  1
  1 + 2x
  1 + 3x +  4x^2
  1 + 4x +  8x^2 +  8x^3
  1 + 5x + 13x^2 + 20x^3 + 16x^4
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  4,  0;
  1, 4,  8,  8,  0;
  1, 5, 13, 20, 16,  0;
  1, 6, 19, 38, 48, 32, 0;
Triangle in A049600 begins:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3,  4;
  0, 1, 4,  8,  8;
  0, 1, 5, 13, 20, 16;
  0, 1, 6, 19, 38, 48, 32;
  ... - Philippe Deléham, Mar 23 2012

Maple

T := (n, k) -> hypergeom([n-k+1, -k], [1], -1):
seq(lprint(seq(simplify(T(n, k)), k=0..n)), n=0..7); # Peter Luschny, May 20 2015

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*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[%]   (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208341 *)

Prog

(Haskell)
a208341 n k = a208341_tabl !! (n-1) !! (k-1)
a208341_row n = a208341_tabl !! (n-1)
a208341_tabl = map reverse a106195_tabl
-- Reinhard Zumkeller, Dec 16 2013
(PARI) T(n, k) = sum(i = 0, k, 2^(k-i)*binomial(n-k, i)*binomial(k, i));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Aug 14 2015

Crossrefs

Cf. A160232, A000045, A049600, A106195.

Sequence in context: A181851 A210231 A180378 * A201634 A210211 A283054

Adjacent sequences: A208338 A208339 A208340 * A208342 A208343 A208344

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 25 2012

Extensions

New name from Peter Luschny, May 20 2015

Offset corrected by Joerg Arndt, Aug 12 2015

Status

approved