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A152534 Triangle T(n,k) read by rows with q-e.g.f.: 1/Product_{k>0} (1-x^k/faq(k,q)).

%I A152534 #33 Feb 16 2025 08:33:09
%S A152534 1,1,2,1,3,3,3,1,5,7,11,11,8,4,1,7,13,25,36,44,42,36,24,13,5,1,11,24,
%T A152534 54,93,142,184,215,222,208,172,126,81,44,19,6,1,15,39,98,195,344,532,
%U A152534 753,964,1150,1264,1294,1226,1082,880,661,451,278,151,70,26,7,1
%N A152534 Triangle T(n,k) read by rows with q-e.g.f.: 1/Product_{k>0} (1-x^k/faq(k,q)).
%H A152534 Alois P. Heinz, <a href="/A152534/b152534.txt">Rows n = 0..50, flattened</a>
%H A152534 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a>.
%H A152534 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A152534 Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} faq(n,q)/Product_{i=1..n} faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2*e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.
%F A152534 Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = 1.
%F A152534 Sum_{k=0..binomial(n,2)} (-1)^k*T(n,k) = A152536(n). - _Alois P. Heinz_, Aug 09 2021
%e A152534 Triangle begins:
%e A152534   1;
%e A152534   1;
%e A152534   2,  1;
%e A152534   3,  3,  3,  1;
%e A152534   5,  7, 11, 11,  8,  4,  1;
%e A152534   7, 13, 25, 36, 44, 42, 36, 24, 13,  5,  1;
%e A152534   ...
%p A152534 multinomial2q := proc(n::integer,k::integer,nparts::integer)
%p A152534         local lpar ,res, constrp;
%p A152534         res := [] ;
%p A152534         if n< 0 or nparts <= 0 then
%p A152534                 ;
%p A152534         elif nparts = 1 then
%p A152534                 if n = k then
%p A152534                         return [[n]] ;
%p A152534                 end if;
%p A152534         else
%p A152534                 for lpar from 0 do
%p A152534                         if lpar*nparts > n or lpar > k then
%p A152534                                 break;
%p A152534                         end if;
%p A152534                         for constrp in procname(n-nparts*lpar,k-lpar,nparts-1) do
%p A152534                                 if nops(constrp) > 0 then
%p A152534                                         res := [op(res),[op(constrp),lpar]] ;
%p A152534                                 end if;
%p A152534                         end do:
%p A152534                 end do:
%p A152534         end if ;
%p A152534         return res ;
%p A152534 end proc:
%p A152534 multinomial2 := proc(n::integer,k::integer)
%p A152534         local res,constrp ;
%p A152534         res := [] ;
%p A152534         for constrp in multinomial2q(n,k,n) do
%p A152534                 if nops(constrp) > 0 then
%p A152534                         res := [op(res),constrp] ;
%p A152534                 end if ;
%p A152534         end do:
%p A152534         res ;
%p A152534 end proc:
%p A152534 faq := proc(i,q)
%p A152534         mul((q^j-1)/(q-1),j=1..i) ;
%p A152534 end proc;
%p A152534 A152534 := proc(n,k)
%p A152534         pi := [] ;
%p A152534         for sp from 0 to n do
%p A152534                 pi := [op(pi),op(multinomial2(n,sp))] ;
%p A152534         end do;
%p A152534         tqk := 0 ;
%p A152534         for p in pi do
%p A152534                 faqe :=1 ;
%p A152534                 for i from 1 to nops(p) do
%p A152534                         faqe := faqe* faq(i,q)^op(i,p) ;
%p A152534                 end do:
%p A152534                 tqk := tqk+faq(n,q)/faqe ;
%p A152534         end do;
%p A152534         tqk ;
%p A152534         coeftayl(tqk,q=0,k) ;
%p A152534 end proc:
%p A152534 for n from 1 to 8 do
%p A152534         for k from 0 to binomial(n,2) do
%p A152534                 printf("%d,",A152534(n,k)) ;
%p A152534         end do;
%p A152534         printf("\n") ;
%p A152534 end do: # _R. J. Mathar_, Sep 27 2011
%p A152534 # second Maple program:
%p A152534 f:= proc(n) option remember; `if`(n<2, 1, f(n-1)*(q^n-1)/(q-1)) end:
%p A152534 b:= proc(n, i) option remember; simplify(`if`(n=0 or i=1, 1,
%p A152534       add(b(n-i*j, i-1)/f(i)^j, j=0..n/i)))
%p A152534     end:
%p A152534 T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(simplify(f(n)*b(n$2))):
%p A152534 seq(T(n), n=0..10);  # _Alois P. Heinz_, Aug 09 2021
%t A152534 f[n_] := f[n] = If[n < 2, 1, f[n - 1]*(q^n - 1)/(q - 1)];
%t A152534 b[n_, i_] := b[n, i] = Simplify[If[n == 0 || i == 1, 1,
%t A152534      Sum[b[n - i*j, i - 1]/f[i]^j, {j, 0, n/i}]]];
%t A152534 T[n_] := CoefficientList[Simplify[f[n]*b[n, n]], q];
%t A152534 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 11 2022, after _Alois P. Heinz_ *)
%Y A152534 Cf. A005651 (row sums), A000041 (first column), A076276 (second column), A152474, A152536.
%Y A152534 T(n,n) gives A346980.
%K A152534 nonn,tabf
%O A152534 0,3
%A A152534 _Vladeta Jovovic_, Dec 06 2008
%E A152534 T(0,0)=1 prepended by _Alois P. Heinz_, Aug 09 2021