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A071947 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

%I A071947 #21 Mar 03 2017 02:45:37
%S A071947 1,1,0,1,1,1,1,2,3,1,1,3,6,6,3,1,4,10,15,15,6,1,5,15,29,40,36,15,1,6,
%T A071947 21,49,84,105,91,36,1,7,28,76,154,238,280,232,91,1,8,36,111,258,468,
%U A071947 672,750,603,232,1,9,45,155,405,837,1398,1890,2025,1585,603,1,10,55,209,605
%N A071947 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.
%H A071947 G. C. Greubel, <a href="/A071947/b071947.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A071947 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.4153/CJM-1997-015-x">On some alternative characterizations of Riordan arrays</a>, Canad J. Math., 49 (1997), 301-320.
%F A071947 G.f.: t*(1+t*z-q)/[(1+t*z)*(2*t^2*z +t*z - 1 + q)], where q = sqrt(1 -2*t*z -3*t^2*z^2).
%F A071947 Sum_{k, 0<=k<=n} T(n,k)*2^(n-k) = A112657(n). - _Philippe Deléham_, Apr 01 2007
%F A071947 T(n,k) = A027907(n,k) - A027907(n,k-1). T(n,n) = A005043(n). # _Peter Luschny_, Oct 01 2014
%e A071947 Triangle begins
%e A071947   1;
%e A071947   1,  0;
%e A071947   1,  1,  1;
%e A071947   1,  2,  3,  1;
%e A071947   1,  3,  6,  6,  3;
%e A071947   1,  4, 10, 15, 15,  6;
%p A071947 A071947_row := proc(n) local G, k; G := expand((1+x+x^2)^n):
%p A071947 seq(coeff(G,x,k) - coeff(G,x,k-1), k=0..n) end:
%p A071947 seq(print(A071947_row(n)), n=0..11); # _Peter Luschny_, Oct 01 2014
%t A071947 A027907[n_, k_] := Sum[Binomial[n, j]*Binomial[j, k - j], {j, 0, n}]; A005043[n_] := Sum[(-1)^k*Binomial[n, k]*Binomial[k, Floor[k/2]], {k, 0, n}]; T[n_, k_] := A027907[n, k] - A027907[n, k - 1]; T[n_, n_] := A005043[n]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, Mar 02 2017 *)
%Y A071947 Row sums give A002426 (central trinomial coefficients). Reversal of A089942.
%Y A071947 Cf. A027907.
%K A071947 nonn,easy,tabl
%O A071947 0,8
%A A071947 _N. J. A. Sloane_, Jun 15 2002
%E A071947 Edited by _Emeric Deutsch_, Mar 04 2004