login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 28 09:35 EST 2014. Contains 250302 sequences.