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A125811 Number of coefficients in the n-th q-Bell number as a polynomial in q. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..56.

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]

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-Fran├žois 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

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

Sequence in context: A175143 A137179 A096777 * A281706 A071424 A008762

Adjacent sequences:  A125808 A125809 A125810 * A125812 A125813 A125814

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 10 2006

EXTENSIONS

More terms from Alois P. Heinz, Aug 04 2009

STATUS

approved

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Last modified August 9 18:32 EDT 2020. Contains 336326 sequences. (Running on oeis4.)