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

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, 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

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..50, flattened

Eric Weisstein's World of Mathematics, q-Exponential Function.

Eric Weisstein's World of Mathematics, q-Factorial.

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.

Sum_{k=0..binomial(n,2)} (-1)^k*T(n,k) = A152536(n). - Alois P. Heinz, Aug 09 2021

Example

Triangle begins:
  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;
  ...

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
# second Maple program:
f:= proc(n) option remember; `if`(n<2, 1, f(n-1)*(q^n-1)/(q-1)) end:
b:= proc(n, i) option remember; simplify(`if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)/f(i)^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(simplify(f(n)*b(n$2))):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 09 2021

Mathematica

f[n_] := f[n] = If[n < 2, 1, f[n - 1]*(q^n - 1)/(q - 1)];
b[n_, i_] := b[n, i] = Simplify[If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]/f[i]^j, {j, 0, n/i}]]];
T[n_] := CoefficientList[Simplify[f[n]*b[n, n]], q];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Crossrefs

Cf. A005651 (row sums), A000041 (first column), A076276 (second column), A152474, A152536.

T(n,n) gives A346980.

Sequence in context: A331855 A178244 A227532 * A136018 A138022 A113278

Adjacent sequences: A152531 A152532 A152533 * A152535 A152536 A152537

Keyword

nonn,tabf

Author

Vladeta Jovovic, Dec 06 2008

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Aug 09 2021

Status

approved