A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 1, 1, 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1, 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1, 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1, 1, 7, 21, 56, 105, 161, 231, 302, 356, 379, 392, 384, 358, 314, 262

Offset

0,6

Comments

Limit of reversed rows equals A126348. Largest term in rows equal A126349.

Links

Alois P. Heinz, Rows n = 0..60, flattened

Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), J. Integer Seqs., Vol. 7, 2004.

Formula

G.f. for row n: B(n,q) = 1/E_q*{0^n + Sum_{k>=1} [(q^k-1)/(q-1)]^n / q-Factorial(k)}, where q-Factorial(k) = Product_{j=1..k} [(q^j-1)/(q-1)] and where E_q = Sum_{n>=0} 1/q-Factorial(n) = Product_{n>=1} (1+(q-1)/q^n).

Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - Alois P. Heinz, Aug 02 2021

Example

Number of terms in row n is: n*(n-1)/2 + 1.
Row functions B(n,q) begin:
B(0,q) = B(1,q) = 1;
B(1,q) = 1 + q;
B(2,q) = 1 + 2*q + q^2 + q^3;
B(3,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
Triangle begins:
1;
1;
1, 1;
1, 2, 1, 1;
1, 3, 3, 4, 2, 1, 1;
1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1; ...

Maple

b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(b(n-1, max(m, j), t+j) , j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Aug 02 2021

Mathematica

B[0, _] = 1; B[n_, q_] := B[n, q] = Sum[Binomial[n-1, k] B[k, q] q^k, {k, 0, n-1}] // Expand; Table[CoefficientList[B[n, q], q], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 08 2016 *)

Prog

(PARI) {B(n, q)=if(n==0, 1, sum(k=0, n-1, binomial(n-1, k)*B(k, q)*q^k))} {T(n, k)=Vec(B(n, q)+O(q^(n*(n-1)/2+1)))[k+1]}
(PARI) /* Alternative formula for the n-th q-Bell number (row n): */ {B(n, q)=local(inf=100); round((0^n + sum(k=1, inf, ((q^k-1)/(q-1))^n/prod(i=1, k, (q^i-1)/(q-1)))) / prod(k=1, inf, 1 + (q-1)/q^k))}

Crossrefs

Row sums give A000110.

Cf. A126348, A126349; factorial variant: A126470.

Cf. A346772.

Sequence in context: A366920 A344678 A079415 * A309240 A057001 A307689

Adjacent sequences: A126344 A126345 A126346 * A126348 A126349 A126350

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 31 2006, May 28 2007

Extensions

Keyword:tabl changed to tabf - R. J. Mathar, Oct 21 2010

Status

approved