OFFSET
0,8
COMMENTS
E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - Paul D. Hanna, Apr 07 2005
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
FORMULA
a(n,k) = Sum{j=0..k} C(k,j) B(j) a(n-1,k-j) where B(j) = C(j,[j/2]) = A001405(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.
E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - Paul D. Hanna, Apr 07 2005
EXAMPLE
Rows start:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 3, 6, 10, 20, ...
1, 2, 6, 18, 60, 200, 700, ...
1, 3, 12, 51, 234, 1110, 5460, ...
1, 4, 20, 108, 624, 3760, 23480, ...
1, 5, 30, 195, 1350, 9770, 73300, ...
1, 6, 42, 318, 2556, 21480, 187140, ...
MAPLE
a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
add(binomial(k, j)*binomial(j, floor(j/2))
*a(n-1, k-j), j=0..k))
end:
seq(seq(a(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, May 06 2014
MATHEMATICA
a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[k, j]*Binomial[j, Floor[j/2]]*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
PROG
(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); k!*polcoeff(polcoeff(1/(1-X*besseli(0, 2*Y)-X*Y*besseli(1, 2*Y)), n, x), k, y)} /* Hanna */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Aug 23 2001
STATUS
approved