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A063967 Triangle with a(n,k) = a(n-1,k) + a(n-2,k) + a(n-1,k-1) + a(n-2,k-1) and a(0,0) = 1. 18
1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 5, 15, 16, 7, 1, 8, 30, 43, 29, 9, 1, 13, 58, 104, 95, 46, 11, 1, 21, 109, 235, 271, 179, 67, 13, 1, 34, 201, 506, 705, 591, 303, 92, 15, 1, 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1, 89, 655, 2123, 3979, 4759, 3780, 2010, 703, 154, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)). The inverse of the signed version (1/(1+x-x^2),x(1-x)/(1+x-x^2)) is abs(A091698). - Paul Barry, Jun 10 2005

Diagonal sums are A002478. - Paul Barry, Nov 09 2005

A026729*A007318 as infinite lower triangular matrices . - Philippe Deléham, Dec 11 2008

Central coefficients T(2n,n) are A137644. - Paul Barry, Apr 15 2010

Product of Riordan arrays (1, x(1+x))*(1/(1-x), x/(1-x)), that is, A026729*A007318. - Paul Barry, Mar 14 2011

Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 12 2011

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

Emanuele Munarini, A generalization of André-Jeannin's symmetric identity, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.

FORMULA

G.f.: 1/(1-x*(1+x)*(1+y)). - Vladeta Jovovic, Oct 11 2003

T(n, k)=sum{j=0..n, C(j, n-j)C(j, k)}. - Paul Barry, Nov 09 2005

Sum_{k, 0<=k<=n}x^k*T(n,k)= (-1)^n*A057086(n), (-1)^n*A057085(n+1), (-1)^n*A057084(n), (-1)^n*A030240(n), (-1)^n*A030192(n), (-1)^n*A030191(n), (-1)^n*A001787(n+1), A000748(n), A108520(n), A049347(n), A000007(n), A000045(n), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n), for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . - Philippe Deléham, Nov 03 2006

EXAMPLE

Rows start (1), (1,1), (2,3,1), (3,7,5,1), etc. a(3,1)=a(2,1)+a(1,1)+a(2,0)+a(1,0)=3+1+2+1=7.

Triangle begins

  1,

  1, 1,

  2, 3, 1,

  3, 7, 5, 1,

  5, 15, 16, 7, 1,

  8, 30, 43, 29, 9, 1,

  13, 58, 104, 95, 46, 11, 1,

  21, 109, 235, 271, 179, 67, 13, 1,

  34, 201, 506, 705, 591, 303, 92, 15, 1

Production matrix of inverse A091698 is

  -1, 1,

  0, -2, 1,

  0, 1, -2, 1,

  0, -1, 1, -2, 1,

  0, 1, -1, 1, -2, 1,

  0, -1, 1, -1, 1, -2, 1,

  0, 1, -1, 1, -1, 1, -2, 1,

  0, -1, 1, -1, 1, -1, 1, -2, 1,

  0, 1, -1, 1, -1, 1, -1, 1, -2, 1

[Paul Barry, Mar 14 2011]

MATHEMATICA

T[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 11 2017, after Paul Barry *)

PROG

(Haskell)

a063967_tabl = [1] : [1, 1] : f [1] [1, 1] where

   f us vs = ws : f vs ws where

     ws = zipWith (+) ([0] ++ us ++ [0]) $

          zipWith (+) (us ++ [0, 0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])

-- Reinhard Zumkeller, Apr 17 2013

CROSSREFS

Columns include A000045 and A023610. Right hand columns include A000012 and A005408. Row sums are A002605.

Matrix inverse: A091698. Matrix square: A091700.

Columns 0-1: A000045(n+1), A023610(n-1). Main diagonal: A000012. a(n, n-1) = A005408(n-1).

Row sums are A002605.

Sequence in context: A188107 A174014 A236376 * A059397 A209567 A208338

Adjacent sequences:  A063964 A063965 A063966 * A063968 A063969 A063970

KEYWORD

easy,nonn,tabl,changed

AUTHOR

Henry Bottomley, Sep 05 2001

STATUS

approved

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Last modified November 13 10:55 EST 2018. Contains 317133 sequences. (Running on oeis4.)