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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). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

LINKS

Table of n, a(n) for n=0..78.

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).

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; ...

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-Fran├žois 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

Cf. A126348, A126349, A000110; factorial variant: A126470.

Sequence in context: A124772 A227543 A079415 * A057001 A173072 A219272

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

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Last modified March 27 12:35 EDT 2017. Contains 284176 sequences.