A171651 - OEIS

%I #19 Dec 16 2023 16:03:37

%S 1,2,1,5,4,1,13,15,6,1,35,52,30,8,1,96,175,130,50,10,1,267,576,525,

%T 260,75,12,1,750,1869,2016,1225,455,105,14,1,2123,6000,7476,5376,2450,

%U 728,140,16,1,6046,19107,27000,22428,12096,4410,1092,180,18,1

%N Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

%H Alois P. Heinz, <a href="/A171651/b171651.txt">Rows n = 0..140, flattened</a>

%F Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively.

%F E.g.f. of column k: exp(x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - _Mélika Tebni_, Dec 16 2023

%e Triangle begins:

%e    1;

%e    2,   1;

%e    5,   4,  1;

%e   13,  15,  6, 1;

%e   35,  52, 30, 8, 1;

%e   ...

%p b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1/2,

%p       expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))

%p       +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n+1$2, 1)):

%p seq(T(n), n=0..12);  # _Alois P. Heinz_, Apr 29 2015

%p # second program:

%p A171651:= (n, k)-> binomial(n,k)*add((-1)^(n-k-j)*binomial(n-k,j)*binomial(2*j+1,j+1),j=0..n-k): seq(print(seq(A171651(n, k), k=0..n)), n=0..9);  # _Mélika Tebni_, Dec 16 2023

%t b[u_, d_, t_] := b[u, d, t] = If[u == 0 && d == 0, 1/2, Expand[If[u == 0, 0, b[u-1, d, 2]*If[t == 3, x, 1]] + If[d == 0, 0, b[u, d-1, If[t == 2, 3, 1]]]]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n+1, n+1, 1] ];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 21 2016, after _Alois P. Heinz_ *)

%Y Cf. A097692.

%Y Cf. A007318, A005773.

%Y Cf. A168491, A099323, A001405, A005773, A001700, A026378, A005573, A122898.

%K nonn,tabl

%O 0,2

%A _Philippe Deléham_, Dec 14 2009

%E Corrected by _Philippe Deléham_, Dec 18 2009