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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)).
4
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
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
KEYWORD
nonn,tabf
AUTHOR
Vladeta Jovovic, Dec 06 2008
EXTENSIONS
T(0,0)=1 prepended by Alois P. Heinz, Aug 09 2021
STATUS
approved