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 A106597 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(n,0)=T(n,n)=1, T(n,k) = T(n-1,k-1) + T(n-1,k) + Sum_{i >= 1} T(n-2i,k-i). 2
 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 27, 27, 9, 1, 1, 11, 44, 72, 44, 11, 1, 1, 13, 65, 149, 149, 65, 13, 1, 1, 15, 90, 266, 388, 266, 90, 15, 1, 1, 17, 119, 431, 836, 836, 431, 119, 17, 1, 1, 19, 152, 652, 1585, 2150, 1585, 652, 152, 19, 1, 1, 21, 189, 937, 2743, 4753, 4753, 2743, 937, 189, 21, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Next term is sum of two terms above you in previous row (as in Pascal's triangle A007318) plus sum of terms directly above you on a vertical line. T(n,n-k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), and (s,s) for s>=1. - Joerg Arndt, Jul 01 2011 Row sums gives A118649. - Emanuele Munarini, Feb 01 2017 LINKS FORMULA G.f.: (1-x^2*y)/(1-x-x*y-2*x^2*y+x^3*y+x^3*y^2).- Emanuele Munarini, Feb 01 2017 EXAMPLE Triangle begins: 1; 1, 1; 1, 3, 1; 1, 5, 5, 1; 1, 7, 14, 7, 1; 1, 9, 27, 27, 9, 1; 1, 11, 44, 72, 44, 11, 1; 1, 13, 65, 149, 149, 65, 13, 1; 1, 15, 90, 266, 388, 266, 90, 15, 1; 1, 17, 119, 431, 836, 836, 431, 119, 17, 1; MATHEMATICA CoefficientList[#, y]& /@ CoefficientList[(1 - x^2 y)/(1 - x - x y - 2x^2 y + x^3 y + x^3 y^2) + O[x]^12, x] // Flatten (* Jean-François Alcover, Oct 30 2018, after Emanuele Munarini *) PROG (PARI) /* same as in A092566, but last line (output) replaced by the following */ /* show as triangle T(n-k, k): */ { for(n=0, N-1, for(k=0, n, print1(T(n-k, k), ", "); ); print(); ); } /* Joerg Arndt, Jul 01 2011 */ CROSSREFS Cf. A118649. Sequence in context: A238339 A302997 A144461 * A108359 A100936 A086620 Adjacent sequences:  A106594 A106595 A106596 * A106598 A106599 A106600 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, May 30 2005 EXTENSIONS More terms from Joshua Zucker, May 10 2006 Definition corrected by Emilie Hogan, Oct 15 2009 STATUS approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)