The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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