A309652
a(n) = [x^n] B(x)^n, where B(x) is g.f. of A000312.
Original entry on oeis.org
1, 1, 9, 106, 1493, 24276, 448122, 9301251, 215547845, 5541171496, 156997349684, 4870353700532, 164366482285898, 5998207807965543, 235388194276592723, 9884482616014596546, 442206843338189113445, 20995082225203329126384, 1054247070579064423466016
Offset: 0
-
B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, B(n),
(h-> add(b(j, h)*b(n-j, i-h), j=0..n))(iquo(i, 2))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 23 2019
-
Table[SeriesCoefficient[(1+Sum[k^k*x^k, {k, 1, n}])^n, {x, 0, n}], {n, 0, 20}]
A326986
G.f.: B(x)*B(x^2)*B(x^3)*..., where B(x) is g.f. of A000312.
Original entry on oeis.org
1, 1, 5, 29, 266, 3163, 46994, 827107, 16828741, 388308078, 10017853262, 285720195351, 8926575094978, 303172417424680, 11121259586618456, 438207141286916539, 18458204444260001120, 827690809585441201775, 39365349178064541861252, 1979267564496263599093676
Offset: 0
-
B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
B(n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 23 2019
-
nmax = 20; CoefficientList[Series[Product[1+Sum[k^k*x^(j*k), {k, 1, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
A309682
G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.
Original entry on oeis.org
1, 1, 4, 10, 33, 81, 282, 762, 2599, 7979, 27343, 89371, 315256, 1078498, 3857048, 13651786, 49475282, 178736186, 655247192, 2401663838, 8883371016, 32906649488, 122619768860, 457836275272, 1716620421629, 6449729802639, 24308647131627, 91800114425437
Offset: 0
-
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1,
C(n), add(C(j)*i^j*b(n-i*j, i-1), j=0..n/i))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Aug 23 2019
-
nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*k*x^k])/(2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Showing 1-3 of 3 results.