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A171651
Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).
4
1, 2, 1, 5, 4, 1, 13, 15, 6, 1, 35, 52, 30, 8, 1, 96, 175, 130, 50, 10, 1, 267, 576, 525, 260, 75, 12, 1, 750, 1869, 2016, 1225, 455, 105, 14, 1, 2123, 6000, 7476, 5376, 2450, 728, 140, 16, 1, 6046, 19107, 27000, 22428, 12096, 4410, 1092, 180, 18, 1
OFFSET
0,2
LINKS
FORMULA
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.
E.g.f. of column k: exp(x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 16 2023
EXAMPLE
Triangle begins:
1;
2, 1;
5, 4, 1;
13, 15, 6, 1;
35, 52, 30, 8, 1;
...
MAPLE
b:= proc(u, d, t) option remember; `if`(u=0 and 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)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n+1$2, 1)):
seq(T(n), n=0..12); # Alois P. Heinz, Apr 29 2015
# Alternative:
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
MATHEMATICA
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[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n+1, n+1, 1] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Dec 14 2009
EXTENSIONS
Corrected by Philippe Deléham, Dec 18 2009
STATUS
approved