A286335
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0
Offset: 0
A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 3, 6, 10, 15, ...
0, 2, 6, 13, 24, 40, ...
0, 2, 9, 24, 51, 95, ...
0, 3, 14, 42, 100, 206, ...
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Aug 29 2019
-
Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
A022600
Expansion of Product_{m>=1} (1+q^m)^(-5).
Original entry on oeis.org
1, -5, 10, -15, 30, -56, 85, -130, 205, -315, 465, -665, 960, -1380, 1925, -2651, 3660, -5020, 6775, -9070, 12126, -16115, 21220, -27765, 36235, -47101, 60810, -78115, 100105, -127825, 162391, -205530, 259475, -326565
Offset: 0
Cf. Related to Expansion of Product_{m>=1} (1+q^m)^k:
A022627 (k=-32),
A022626 (k=-31),
A022625 (k=-30),
A022624 (k=-29),
A022623 (k=-28),
A022622 (k=-27),
A022621 (k=-26),
A022620 (k=-25),
A007191 (k=-24),
A022618 (k=-23),
A022617 (k=-22),
A022616 (k=-21),
A022615 (k=-20),
A022614 (k=-19),
A022613 (k=-18),
A022612 (k=-17),
A022611 (k=-16),
A022610 (k=-15),
A022609 (k=-14),
A022608 (k=-13),
A007249 (k=-12),
A022606 (k=-11),
A022605 (k=-10),
A022604 (k=-9),
A007259 (k=-8),
A022602 (k=-7),
A022601 (k=-6), this sequence (k=-5),
A022599 (k=-4),
A022598 (k=-3),
A022597 (k=-2),
A081362 (k=-1),
A000009 (k=1),
A022567 (k=2),
A022568 (k=3),
A022569 (k=4),
A022570 (k=5),
A022571 (k=6),
A022572 (k=7),
A022573 (k=8),
A022574 (k=9),
A022575 (k=10),
A022576 (k=11),
A022577 (k=12),
A022578 (k=13),
A022579 (k=14),
A022580 (k=15),
A022581 (k=16),
A022582 (k=17),
A022583 (k=18),
A022584 (k=19),
A022585 (k=20),
A022586 (k=21),
A022587 (k=22),
A022588 (k=23),
A014103 (k=24),
A022589 (k=25),
A022590 (k=26),
A022591 (k=27),
A022592 (k=28),
A022593 (k=29),
A022594 (k=30),
A022595 (k=31),
A022596 (k=32),
A025233 (k=48).
-
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
-
x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017
A025233
Expansion of Product_{m>=1} (1 + q^m)^48.
Original entry on oeis.org
1, 48, 1176, 19648, 252204, 2655456, 23901760, 189208704, 1344644814, 8713158928, 52107076128, 290374290624, 1519725061816, 7518508799904, 35352238216704, 158716136933504, 683059486979301, 2827559773199856
Offset: 0
-
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^48, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
Showing 1-3 of 3 results.
Comments