A264686
Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k).
Original entry on oeis.org
1, 3, 6, 15, 27, 51, 93, 159, 264, 432, 696, 1086, 1683, 2553, 3837, 5700, 8367, 12147, 17505, 24972, 35361, 49728, 69402, 96243, 132657, 181782, 247692, 335838, 453042, 608289, 813102, 1082256, 1434519, 1894215, 2491644, 3265869, 4265973, 5553771, 7207167
Offset: 0
-
b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
end:
a:= n-> add(b(i$2)*combinat[numbpart](n-i), i=0..n):
seq(a(n), n=0..60); # Alois P. Heinz, Dec 22 2017
-
nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
-
{ my(n=40); Vec(prod(k=1, n, 3/(1-x^k) - 2 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017
A321884
Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 8, 5, 0, 1, 5, 8, 15, 14, 7, 0, 1, 6, 10, 24, 27, 24, 11, 0, 1, 7, 12, 35, 44, 51, 40, 15, 0, 1, 8, 14, 48, 65, 88, 93, 64, 22, 0, 1, 9, 16, 63, 90, 135, 176, 159, 100, 30, 0, 1, 10, 18, 80, 119, 192, 295, 312, 264, 154, 42, 0
Offset: 0
A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, ...
0, 2, 4, 6, 8, 10, 12, 14, 16, ...
0, 3, 8, 15, 24, 35, 48, 63, 80, ...
0, 5, 14, 27, 44, 65, 90, 119, 152, ...
0, 7, 24, 51, 88, 135, 192, 259, 336, ...
0, 11, 40, 93, 176, 295, 456, 665, 928, ...
0, 15, 64, 159, 312, 535, 840, 1239, 1744, ...
0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)
A268499
Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).
Original entry on oeis.org
1, 2, 0, 8, -2, 8, 16, 8, 8, 10, 80, -8, 72, -24, 144, 128, 134, 40, 224, 120, 232, 688, 176, 696, 32, 1194, -96, 1840, 1144, 2248, 288, 2968, 800, 4160, 752, 5104, 6438, 4984, 5104, 5488, 10960, 4856, 14080, 3480, 24408, 15448, 26832, 7080, 42120, 11178
Offset: 0
-
nmax = 100; CoefficientList[Series[Product[(1+3*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Showing 1-3 of 3 results.
Comments