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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

Table of n, a(n) for n=1..63.

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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Last modified December 9 17:46 EST 2016. Contains 278985 sequences.