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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

%I

%S 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,

%T 29,26,20,14,12,7,4,2,1,1,1,6,15,35,55,71,90,101,100,89,82,68,53,38,

%U 26,20,12,7,4,2,1,1,1,7,21,56,105,161,231,302,356,379,392,384,358,314,262

%N 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).

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

%H Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Wagner/wagner3.pdf">Partition Statistics and q-Bell Numbers (q = -1)</a>, J. Integer Seqs., Vol. 7, 2004.

%F 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).

%e Number of terms in row n is: n*(n-1)/2 + 1.

%e Row functions B(n,q) begin:

%e B(0,q) = B(1,q) = 1;

%e B(1,q) = 1 + q;

%e B(2,q) = 1 + 2*q + q^2 + q^3;

%e B(3,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.

%e Triangle begins:

%e 1;

%e 1;

%e 1, 1;

%e 1, 2, 1, 1;

%e 1, 3, 3, 4, 2, 1, 1;

%e 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;

%e 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;

%e 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1; ...

%t 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 *)

%o (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]}

%o (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))}

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

%K nonn,tabf

%O 0,6

%A _Paul D. Hanna_, Dec 31 2006, May 28 2007

%E Keyword:tabl changed to tabf - _R. J. Mathar_, Oct 21 2010

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Last modified April 11 03:42 EDT 2021. Contains 342886 sequences. (Running on oeis4.)