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A238941 Triangle T(n,k), read by rows given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A238941 #10 Mar 14 2017 11:13:02
%S A238941 1,1,1,2,3,1,5,8,4,1,13,21,13,6,1,34,55,40,25,7,1,89,144,120,90,33,9,
%T A238941 1,233,377,354,300,132,51,10,1,610,987,1031,954,483,234,62,12,1,1597,
%U A238941 2584,2972,2939,1671,951,308,86,13,1,4181,6765,8495,8850,5561,3573,1345,480,100,15,1
%N A238941 Triangle T(n,k), read by rows given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A238941 Row sums are A025192(n).
%H A238941 Indranil Ghosh, <a href="/A238941/b238941.txt">Rows 0..100, flattened</a>
%F A238941 G.f. for the column k: x^k*(1-2*x)^A059841(k)/(1-3*x+x^2)^A008619(k).
%F A238941 G.f.: (1-2*x+x*y)/(1-3*x+x^2-x^2*y^2).
%F A238941 T(n,k) = 3*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
%F A238941 Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A001519(n), A025192(n), A030195(n+1) for x = -1, 0, 1, 2 respectively.
%F A238941 Sum_{k = 0..n} T(n,k)*3^k = A015525(n) + A015525(n+1).
%e A238941 Triangle begins:
%e A238941 1;
%e A238941 1, 1;
%e A238941 2, 3, 1;
%e A238941 5, 8, 4, 1;
%e A238941 13, 21, 13, 6, 1;
%e A238941 34, 55, 40, 25, 7, 1;
%e A238941 89, 144, 120, 90, 33, 9, 1;
%e A238941 233, 377, 354, 300, 132, 51, 10, 1;
%t A238941 nmax=10; Column[CoefficientList[Series[CoefficientList[Series[(1 - 2*x + x*y)/(1 - 3*x + x^2 - x^2*y^2), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* _Indranil Ghosh_, Mar 14 2017 *)
%Y A238941 Cf. Columns: A001519, A001906, A238846, A001871.
%Y A238941 Cf. Diagonals: A000012, A032766.
%K A238941 nonn,tabl
%O A238941 0,4
%A A238941 _Philippe Deléham_, Mar 07 2014
%E A238941 Data section corrected and extended by _Indranil Ghosh_, Mar 14 2017