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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).

%I A126347 #30 Jul 11 2025 17:43:21
%S A126347 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 A126347 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 A126347 26,20,12,7,4,2,1,1,1,7,21,56,105,161,231,302,356,379,392,384,358,314,262
%N 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).
%C A126347 Limit of reversed rows equals A126348. Largest term in rows equal A126349.
%H A126347 Alois P. Heinz, <a href="/A126347/b126347.txt">Rows n = 0..60, flattened</a>
%H A126347 Carl G. Wagner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Wagner/wagner3.html">Partition Statistics and q-Bell Numbers (q = -1)</a>, J. Integer Seqs., Vol. 7, 2004.
%F A126347 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).
%F A126347 Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - _Alois P. Heinz_, Aug 02 2021
%F A126347 Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - _Mikhail Kurkov_, Jul 06 2025
%e A126347 Number of terms in row n is: n*(n-1)/2 + 1.
%e A126347 Row functions B(n,q) begin:
%e A126347   B(0,q) = 1;
%e A126347   B(1,q) = 1;
%e A126347   B(2,q) = 1 + q;
%e A126347   B(3,q) = 1 + 2*q + q^2 + q^3;
%e A126347   B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
%e A126347 Triangle begins:
%e A126347   1;
%e A126347   1;
%e A126347   1, 1;
%e A126347   1, 2, 1, 1;
%e A126347   1, 3, 3, 4, 2, 1, 1;
%e A126347   1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
%e A126347   1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
%e A126347   1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1;
%e A126347   ...
%p A126347 b:= proc(n, m, t) option remember; `if`(n=0, x^t,
%p A126347       add(b(n-1, max(m, j), t+j) , j=1..m+1))
%p A126347     end:
%p A126347 T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)):
%p A126347 seq(T(n), n=0..8);  # _Alois P. Heinz_, Aug 02 2021
%t A126347 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 *)
%o A126347 (PARI) {B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
%o A126347 row(n)={Vec(B(n, 'q)+O('q^(n*(n-1)/2+1)))}
%o A126347 (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 A126347 Row sums give A000110.
%Y A126347 Cf. A126348, A126349; factorial variant: A126470.
%Y A126347 Cf. A346772.
%K A126347 nonn,tabf
%O A126347 0,6
%A A126347 _Paul D. Hanna_, Dec 31 2006, May 28 2007
%E A126347 Keyword:tabl changed to tabf by _R. J. Mathar_, Oct 21 2010