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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. 19
 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 *) (* Function RiordanSquare defined in A321620. *) RiordanSquare[1/(1 - x - x^2), 11] // Flatten (* Peter Luschny, Nov 27 2018 *) 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 Row sums are A002605. Columns include: A000045(n+1), A023610(n-1). Main diagonal: A000012, a(n, n-1) = A005408(n-1). Matrix inverse: A091698, matrix square: A091700. Cf. A321620. Sequence in context: A188107 A174014 A236376 * A059397 A209567 A208338 Adjacent sequences:  A063964 A063965 A063966 * A063968 A063969 A063970 KEYWORD easy,nonn,tabl AUTHOR Henry Bottomley, Sep 05 2001 STATUS approved

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Last modified October 21 08:16 EDT 2019. Contains 328292 sequences. (Running on oeis4.)