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A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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)

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, Joerg Arndt, Aug 12 2015

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)