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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)). 2
 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, 54, 93, 142, 184, 215, 222, 208, 172, 126, 81, 44, 19, 6, 1, 15, 39, 98, 195, 344, 532, 753, 964, 1150, 1264, 1294, 1226, 1082, 880, 661, 451, 278, 151, 70, 26, 7, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Eric Weisstein, q-Exponential Function from MathWorld. Eric Weisstein, q-Factorial from MathWorld. FORMULA 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. Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n)) = 1. EXAMPLE 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; MAPLE multinomial2q := proc(n::integer, k::integer, nparts::integer)         local lpar , res, constrp;         res := [] ;         if n< 0 or nparts <= 0 then                 ;         elif nparts = 1 then                 if n = k then                         return [[n]] ;                 end if;         else                 for lpar from 0 do                         if lpar*nparts > n or lpar > k then                                 break;                         end if;                         for constrp in procname(n-nparts*lpar, k-lpar, nparts-1) do                                 if nops(constrp) > 0 then                                         res := [op(res), [op(constrp), lpar]] ;                                 end if;                         end do:                 end do:         end if ;         return res ; end proc: multinomial2 := proc(n::integer, k::integer)         local res, constrp ;         res := [] ;         for constrp in multinomial2q(n, k, n) do                 if nops(constrp) > 0 then                         res := [op(res), constrp] ;                 end if ;         end do:         res ; end proc: faq := proc(i, q)         mul((q^j-1)/(q-1), j=1..i) ; end proc; A152534 := proc(n, k)         pi := [] ;         for sp from 0 to n do                 pi := [op(pi), op(multinomial2(n, sp))] ;         end do;         tqk := 0 ;         for p in pi do                 faqe :=1 ;                 for i from 1 to nops(p) do                         faqe := faqe* faq(i, q)^op(i, p) ;                 end do:                 tqk := tqk+faq(n, q)/faqe ;         end do;         tqk ;         coeftayl(tqk, q=0, k) ; end proc: for n from 1 to 8 do         for k from 0 to binomial(n, 2) do                 printf("%d, ", A152534(n, k)) ;         end do;         printf("\n") ; end do: # R. J. Mathar, Sep 27 2011 CROSSREFS Cf. A005651(row sums), A000041(first column), A076276(second column), A152474. Sequence in context: A039775 A178244 A227532 * A136018 A138022 A113278 Adjacent sequences:  A152531 A152532 A152533 * A152535 A152536 A152537 KEYWORD nonn,tabf AUTHOR Vladeta Jovovic, Dec 06 2008 STATUS approved

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