0,2

Reverse of A029635. Row sums are A003945. Diagonal sums are Fib(n+2) = sum_{k=0..floor(n/2)} (2n-3k)C(n-k,n-2k)/(n-k). - Paul Barry, Jan 30 2005

Riordan array ((1+x)/(1-x), x/(1-x)). The signed triangle (-1)^(n-k)T(n,k) or ((1-x)/(1+x), x/(1+x)) is the inverse of A055248. Row sums are A003945. Diagonal sums are F(n+2). - Paul Barry, Feb 03 2005

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96...) = (1, 3, 7, 15, 31, 63, 127...) - (0, 0, 1, 3, 7, 15, 31,...); where (1, 3, 7, 15,...) = A000225. - Gary W. Adamson, Apr 22 2007

Triangle T(n,k), read by rows, given by (2,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 17 2011

A029653 is jointly generated with A208510 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*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012

For a closed-form formula for arbitrary left and right borders of Pascal like triangle, see A228196. - Boris Putievskiy, Aug 18 2013

For a closed-form formula for generalized Pascal's triangle, see A228576. - Boris Putievskiy, Sep 04 2013

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 39.

H. Hosoya, Pascal's triangle, non-adjacent numbers and D-dimensional atomic orbitals, J. Math. Chemistry, vol. 23, 1998, 169-178.

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

Mark C. Wilson, Asymptotics for generalized Riordan arrays. International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005.

T(n, k) = C(n-2, k-1) + C(n-2, k) + C(n-1, k-1) + C(n-1, k).

G.f.: (1 + x + y + xy)/(1 - y - xy). - Ralf Stephan, May 17 2004

T(n, k) = (2n-k)*binomial(n, n-k)/n, n, k>0. - Paul Barry, Jan 30 2005

Sum_{k=0..n} T(n, k)*x^k gives A003945-A003954 for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - Philippe Deléham, Jul 10 2005

T(n, k) = C(n-1, k) + C(n, k) . - Philippe Deléham, Jul 10 2005

Equals A097806 * A007318, i.e., the pairwise operator * Pascal's Triangle as infinite lower triangular matrices. - Gary W. Adamson, Apr 22 2007

From Peter Bala, Dec 27 2014: (Start)

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2 + 5*x + 4*x^2/2! + x^3/3!) = 2 + 7*x + 16*x^2/2! + 30*x^3/3! + 50*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).

Let M denote the lower unit triangular array with 1's on the main diagonal and 1's everywhere else below the main diagonal except for the first column which consists of the sequence [1,2,2,2,....]. For k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

The triangle T(n,k) begins:

n\k 0  1  2   3   4   5   6   7  8  9 10 ...

0:  1

1:  2  1

2:  2  3  1

3:  2  5  4   1

4:  2  7  9   5   1

5:  2  9 16  14   6   1

6:  2 11 25  30  20   7   1

7:  2 13 36  55  50  27   8   1

8:  2 15 49  91 105  77  35   9  1

9:  2 17 64 140 196 182 112  44 10  1

10: 2 19 81 204 336 378 294 156 54 11  1

... Reformatted. - Wolfdieter Lang, Jan 09 2015

With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins

/1        \/1         \/1        \      /1        \

|2 1      ||0 1       ||0 1      |      |2 1      |

|2 1 1    ||0 2 1     ||0 0 1    |... = |2 3 1    |

|2 1 1 1  ||0 2 1 1   ||0 0 2 1  |      |2 5 4 1  |

|2 1 1 1 1||0 2 1 1 1 ||0 0 2 1 1|      |2 7 9 5 1|

|...      ||...       ||...      |      |...      |

- Peter Bala, Dec 27 2014

A029653 :=  proc(n, k)

if n = 0 then

  1;

else

  binomial(n-1, k)+binomial(n, k)

fi

end proc: # R. J. Mathar, Jun 30 2013

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;

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[%]  (* A208510 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]  (* A029653 *)

(* Clark Kimberling, Feb 28 2012 *)

(Haskell)

a029653 n k = a029653_tabl !! n !! k

a029653_row n = a029653_tabl !! n

a029653_tabl = [1] : iterate

               (\xs -> zipWith (+) ([0] ++ xs) (xs ++ [0])) [2, 1]

-- Reinhard Zumkeller, Dec 16 2013

(Python)

from sympy import Poly

def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)

def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1

def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]

for n in xrange(1, 13): print a(n) # Indranil Ghosh, May 27 2017

(d, 1) Pascal triangles for d=3..10: A093560-5, A093644-5.

Cf. A007318, A003945, A208510, A228196, A228576.

Cf. A078812, A106195.

Sequence in context: A065158 A181842 A209564 * A067763 A263683 A087730

Adjacent sequences:  A029650 A029651 A029652 * A029654 A029655 A029656

nonn,tabl,changed

Mohammad K. Azarian

More terms from James A. Sellers

approved