A152474 -id:A152474 - cp's OEIS frontend

A346981 a(n) = A152474(n,n).

1, 0, 0, 1, 41, 842, 20520, 477479, 12482191, 344597977, 10325683780, 329996493091, 11307950123833, 411428962250775, 15890609817681079, 648195555340597125, 27864181100124570327, 1258096888119566215689, 59531788666265363070393, 2944807922604446013781174

Offset: 0

Alois P. Heinz, Aug 09 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Main diagonal of A152474.

Cf. A346980.

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, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)/f(i)^j/j!, j=0..n/i))))
    end:
a:= n-> coeff(simplify(n!*f(n)*b(n$2)), q, n):
seq(a(n), n=0..19);

f[n_] := f[n] = If[n < 2, 1, f[n - 1]*(q^n - 1)/(q - 1)];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1]/f[i]^j/j!, {j, 0, n/i}]]];
a[n_] := SeriesCoefficient[n!*f[n]*b[n, n], {q, 0, n}];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Apr 07 2022, after Alois P. Heinz *)

a(n) = A152474(n,n).

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

Original entry on oeis.org

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

Vladeta Jovovic, Dec 06 2008

			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;
  ...

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.

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

T(n,n) gives A346980.

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

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 *)

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

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

cp's OEIS Frontend

A346981 a(n) = A152474(n,n).

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