A341394
Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^10.
Original entry on oeis.org
1, 20, 230, 1940, 13285, 77944, 405250, 1910330, 8300380, 33655860, 128574734, 466317760, 1615509765, 5373215450, 17230062315, 53457917856, 160963157005, 471587847690, 1347417640405, 3761860656610, 10280578499844, 27543107112940, 72440412567485
Offset: 10
Cf.
A026007,
A321955,
A327388,
A341384,
A341385,
A341386,
A341387,
A341388,
A341390,
A341391,
A341393.
-
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, 10):
seq(a(n), n=10..32); # Alois P. Heinz, Feb 10 2021
-
nmax = 32; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &
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.