A125811 Number of coefficients in the n-th q-Bell number as a polynomial in q.

1, 1, 1, 2, 3, 5, 8, 11, 15, 20, 26, 32, 39, 47, 56, 66, 76, 87, 99, 112, 126, 141, 156, 172, 189, 207, 226, 246, 267, 288, 310, 333, 357, 382, 408, 435, 463, 491, 520, 550, 581, 613, 646, 680, 715, 751, 787, 824, 862, 901, 941, 982, 1024, 1067, 1111, 1156, 1201

Offset: 0

Author

Paul D. Hanna, Dec 10 2006

Keywords

• nonn

Examples

			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.
		

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.

Crossrefs

Cf. A125810, A125812, A125813, A125814, A125815.

Programs

Maple

Cq:= proc(n,k) local j; if 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 *)

PARI

/* q-Binomial coefficients: */
C_q(n,k)=if(n
				

Python

from math import comb, isqrt
def A125811(n): return 1+comb(n,2)-sum(isqrt((k<<3)+1)-1>>1 for k in range(1,n)) # Chai Wah Wu, Feb 27 2025

Formula

a(n) = A023536(n-2) + 1.

a(n) = n*(n+1)/2 - 4 - Sum_{k=2..n-2} floor(1/2 + sqrt(2*k+4)) for n>2. [Due to a formula by Jan Hagberg in A023536]

Extensions

More terms from Alois P. Heinz, Aug 04 2009