OFFSET
0,4
LINKS
Arvind Ayyer and Naren Sundaravaradan, An area-bounce exchanging bijection on a large subset of Dyck paths, arXiv:2401.14668 [math.CO], 2024. See p. 20.
FORMULA
EXAMPLE
This sequence gives the number of terms in rows of A125810.
Row g.f.s B_q(n) of A125810 are polynomials in q generated by:
B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0 with B_q(0)=1
where the triangle of q-binomial coefficients C_q(n,k) begins:
1;
1, 1;
1, 1 + q, 1;
1, 1 + q + q^2, 1 + q + q^2, 1;
1, 1 + q + q^2 + q^3, 1 + q + 2*q^2 + q^3 + q^4, 1 + q + q^2 + q^3, 1;
The initial q-Bell coefficients in B_q(n) are:
B_q(0) = 1; B_q(1) = 1; B_q(2) = 2;
B_q(3) = 4 + q;
B_q(4) = 8 + 4*q + 3*q^2;
B_q(5) = 16 + 12*q + 13*q^2 + 8*q^3 + 3*q^4;
B_q(6) = 32 + 32*q + 42*q^2 + 38*q^3 + 33*q^4 + 15*q^5 + 10*q^6 + q^7.
MAPLE
Cq:= proc(n, k) local j; if n<k or k<0 then 0 elif n=0 or k=0 then 1 else mul(1-q^j, j=n-k+1..n)/mul(1-q^j, j=1..k) fi end: Bq:= proc(n) option remember; local k; if n=0 then 1 else simplify(add(Bq(k) * Cq(n-1, k), k=0..n-1)) fi end: a:= n-> nops(Bq(n)): seq(a(n), n=0..60); # Alois P. Heinz, Aug 04 2009
MATHEMATICA
QB[n_, q_] := QB[n, q] = Sum[QB[j, q] QBinomial[n-1, j, q], {j, 0, n-1}] // FunctionExpand // Simplify; QB[0, q_]=1; QB[1, q_]=1; a[n_] := CoefficientList[QB[n, q], q] // Length; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Feb 29 2016 *)
PROG
(PARI) /* q-Binomial coefficients: */
C_q(n, k)=if(n<k || k<0, 0, if(n==0 || k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j)))
/* q-Bell numbers = eigensequence of q-binomial triangle: */
B_q(n)=if(n==0, 1, sum(k=0, n-1, B_q(k)*C_q(n-1, k)))
/* Number of coefficients in B_q(n) as a polynomial in q: */
a(n)=#Vec(B_q(n))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 10 2006
EXTENSIONS
More terms from Alois P. Heinz, Aug 04 2009
STATUS
approved