A341393
Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^9.
Original entry on oeis.org
1, 18, 189, 1464, 9252, 50292, 243117, 1068939, 4344660, 16522967, 59349627, 202844007, 663615180, 2088375867, 6347592999, 18698498610, 53538715836, 149375490453, 406987481852, 1084906793142, 2834211905622, 7266665613438, 18308976116535
Offset: 9
Cf.
A026007,
A321954,
A327387,
A341384,
A341385,
A341386,
A341387,
A341388,
A341390,
A341391,
A341394.
-
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
`if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 9):
seq(a(n), n=9..31); # Alois P. Heinz, Feb 10 2021
-
nmax = 31; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &
A341395
Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.
Original entry on oeis.org
1, 2, 14, 92, 662, 4872, 36578, 278161, 2135902, 16522967, 128574734, 1005321616, 7891885382, 62160038813, 491003317483, 3888045701232, 30854283708670, 245315312649653, 1953735732991919, 15583347966328833, 124463844976490422, 995305632560023009, 7968042676400949882
Offset: 0
Cf.
A026007,
A257675,
A270913,
A270922,
A324595,
A341384,
A341385,
A341386,
A341387,
A341388,
A341390,
A341391,
A341393,
A341394.
-
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
`if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k<2, g(n+1), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..22); # Alois P. Heinz, Feb 10 2021
-
Join[{1}, Table[SeriesCoefficient[(-1 + Product[(1 + x^k)^k, {k, 1, 2 n}])^n, {x, 0, 2 n}], {n, 1, 22}]]
A[n_, k_] := A[n, k] = If[n == 0, 1, k Sum[A[n - j, k] Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 22}]
Showing 1-2 of 2 results.