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A091533 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,1.

%I A091533 #38 Apr 26 2025 07:57:15
%S A091533 1,1,1,2,3,2,3,7,7,3,5,15,21,15,5,8,30,53,53,30,8,13,58,124,157,124,
%T A091533 58,13,21,109,273,417,417,273,109,21,34,201,577,1029,1239,1029,577,
%U A091533 201,34,55,365,1181,2405,3375,3375,2405,1181,365,55,89,655,2358,5393,8625,10047,8625,5393,2358,655,89
%N A091533 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,1.
%C A091533 T(n,k) is the number of lattice paths from (0,0) to (k,n-k) using steps (1,0),(2,0),(0,1),(0,2),(1,1). - _Seiichi Manyama_, Apr 26 2025.
%H A091533 Seiichi Manyama, <a href="/A091533/b091533.txt">Rows n = 0..139, flattened</a>
%F A091533 T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.
%F A091533 G.f.: A(x, y) = 1/(1-x-x*y-x^2-x^2*y-x^2*y^2).
%F A091533 Sum_{k = 0..n} T(n,k)*x^k = A000045(n+1), A015518(n+1), A015524(n+1), A200069(n+1) for x = 0, 1, 2, 3 respectively. - _Philippe Deléham_, Oct 30 2013
%F A091533 Sum_{k = 0..floor(n/2)} T(n-k,k) = (-1)^n*A079926(n). - _Philippe Deléham_, Oct 30 2013
%e A091533 This triangle begins:
%e A091533    1;
%e A091533    1,   1;
%e A091533    2,   3,    2;
%e A091533    3,   7,    7,    3;
%e A091533    5,  15,   21,   15,    5;
%e A091533    8,  30,   53,   53,   30,     8;
%e A091533   13,  58,  124,  157,  124,    58,   13;
%e A091533   21, 109,  273,  417,  417,   273,  109,   21;
%e A091533   34, 201,  577, 1029, 1239,  1029,  577,  201,   34;
%e A091533   55, 365, 1181, 2405, 3375,  3375, 2405, 1181,  365,  55;
%e A091533   89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89;
%e A091533   ...
%p A091533 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A091533      `if`(n<1, 1, add(add(T(n-i, k-j), j=0..i), i=1..2)))
%p A091533     end:
%p A091533 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jan 14 2022
%t A091533 A091533[-2, n2_] = 0; A091533[n1_, -2] = 0; A091533[-1, n2_] = 0; A091533[n1_, -1] = 0; A091533[0, 0] = 1; A091533[n1_, n2_] := A091533[n1, n2] = A091533[n1 - 1, n2] + A091533[n1, n2 - 1] + A091533[n1 - 1, n2 - 1] + A091533[n1 - 2, n2] + A091533[n1, n2 - 2]; Table[A091533[x - y, y], {x, 0, 9}, {y, 0, x}] // Flatten (* _Robert P. P. McKone_, Jan 14 2022 *)
%Y A091533 Row sums: A015518(n+1). Columns 0-1: A000045(n+1), A023610(n-1).
%Y A091533 Cf. A090174, A212338 (column 2), A192364 (central terms).
%Y A091533 Cf. A036355.
%K A091533 nonn,easy,tabl
%O A091533 0,4
%A A091533 _Christian G. Bower_, Jan 19 2004