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Triangle read by rows, related to Pascal's triangle.
10

%I #30 Jan 15 2022 15:29:18

%S 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 58,13,21,109,273,417,417,273,109,21,34,201,577,1029,1239,1029,577,

%U 201,34,55,365,1181,2405,3375,3375,2405,1181,365,55,89,655,2358,5393,8625,10047,8625,5393,2358,655,89

%N Triangle read by rows, related to Pascal's triangle.

%H Seiichi Manyama, <a href="/A091533/b091533.txt">Rows n = 0..139, flattened</a>

%F 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 G.f.: A(x, y) = 1/(1-x-x*y-x^2-x^2*y-x^2*y^2).

%F 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 Sum_{k = 0..floor(n/2)} T(n-k,k) = (-1)^n*A079926(n). - _Philippe Deléham_, Oct 30 2013

%e This triangle begins:

%e 1;

%e 1, 1;

%e 2, 3, 2;

%e 3, 7, 7, 3;

%e 5, 15, 21, 15, 5;

%e 8, 30, 53, 53, 30, 8;

%e 13, 58, 124, 157, 124, 58, 13;

%e 21, 109, 273, 417, 417, 273, 109, 21;

%e 34, 201, 577, 1029, 1239, 1029, 577, 201, 34;

%e 55, 365, 1181, 2405, 3375, 3375, 2405, 1181, 365, 55;

%e 89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89;

%e ...

%p T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,

%p `if`(n<1, 1, add(add(T(n-i, k-j), j=0..i), i=1..2)))

%p end:

%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Jan 14 2022

%t 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 Row sums: A015518(n+1). Columns 0-1: A000045(n+1), A023610(n-1).

%Y Cf. A090174, A212338 (column 2), A192364 (central terms).

%K nonn,easy,tabl

%O 0,4

%A _Christian G. Bower_, Jan 19 2004