A107742 -id:A107742 - cp's OEIS frontend

A333653 Expansion of Product_{i>=1, j>=1} (1 + ix^(ij)).

1, 1, 3, 7, 13, 27, 54, 98, 174, 335, 572, 1004, 1733, 2933, 4916, 8307, 13470, 22042, 35851, 57256, 91462, 145231, 227667, 355522, 554058, 853986, 1313121, 2010318, 3057827, 4627213, 6989808, 10481205, 15679549, 23365207, 34658909, 51241077, 75541695, 110852295, 162238415

Offset: 0

Seiichi Manyama, Aug 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A022629, A107742, A285244, A318416, A332199.

m = 38; CoefficientList[Series[Product[1 + i*x^(i*j), {i, 1, m}, {j, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 23 2020 *)

N=40; x='x+O('x^N); Vec(prod(i=1, N, prod(j=1, N\i, 1+i*x^(i*j))))

N=40; x='x+O('x^N); Vec(prod(k=1, N, prod(d=1, k, 1+(k%d==0)*d*x^k)))

G.f.: Product_{k>0} f(q^k) where f(q) = Product_{i>=1} (1 + i*q^i).

G.f.: Product_{k>0} Product_{d|k} (1 + d*x^k).

A356957 Number of set partitions of strict integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1.

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 7, 7, 8, 13, 20, 19, 27, 30, 42, 60, 63, 75, 99, 112, 141, 191, 205, 248, 296, 357, 408, 513, 617, 696, 831, 969, 1117, 1337, 1523, 1797, 2171, 2420, 2805, 3265, 3772, 4289, 5013, 5661, 6579, 7679, 8615, 9807, 11335, 12799, 14581

Offset: 0

Gus Wiseman, Sep 13 2022

nonn

			The a(1) = 1 through a(6) = 7 set partitions:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1},{3}}  {{2,3}}    {{1,2,3}}
                {{1},{2}}             {{1},{4}}  {{1},{5}}
                                      {{2},{3}}  {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{1,2},{3}}
                                                 {{1},{2},{3}}

Intervals are counted by A000012, A001227, ranked by A073485.
The initial version is A010054.
For set partitions of {1..n} we have A011782.
The non-strict version is A107742
Not restricting to intervals gives A294617.
More types: A356936, A356937, A356938, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A001970 counts multiset partitions of integer partitions.
A356941 counts multiset partitions of integer partitions w/ gapless blocks.
Cf. A001055, A055887, A061260, A270995, A304969, A349050, A349055.

chQ[y_] := Length[y] <= 1 || Union[Differences[y]] == {1};
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@sps/@Reverse/@Select[IntegerPartitions[n], UnsameQ@@#&],And@@chQ/@#&]],{n,0,15}]

A318844 Expansion of Product_{k>=1} (1 + x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 8, 10, 15, 17, 29, 31, 48, 60, 81, 99, 143, 167, 231, 287, 374, 460, 615, 740, 964, 1194, 1512, 1856, 2379, 2877, 3635, 4460, 5540, 6759, 8433, 10192, 12608, 15335, 18774, 22726, 27868, 33525, 40863, 49292, 59652, 71694, 86780, 103818, 125118, 149778, 179608

Offset: 0

Ilya Gutkovskiy, Sep 04 2018

nonn

Convolution of A081362 and A107742.

Weigh transform of A032741.

N. J. A. Sloane, Transforms

Cf. A000005, A000203, A032741, A081362, A107742, A318783.

with(numtheory): a:=series(mul((1+x^k)^(tau(k)-1),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

nmax = 52; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 1) x^k/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

G.f.: Product_{k>=1} (1 + x^k)^A032741(k).

G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/(k*(1 - x^(2*k)))), where sigma_1(k) = sum of divisors of k (A000203).

A327746 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i(2j - 1))).

Original entry on oeis.org

1, 1, 0, 2, 2, 2, 3, 3, 6, 7, 8, 9, 14, 16, 17, 26, 30, 35, 43, 52, 62, 77, 87, 104, 133, 152, 173, 212, 251, 287, 344, 397, 465, 549, 627, 729, 864, 986, 1127, 1325, 1524, 1740, 2009, 2306, 2641, 3047, 3455, 3942, 4549, 5157, 5846, 6700, 7605, 8608

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Jason Bard, Table of n, a(n) for n = 0..2000

Cf. A001227, A107742, A288007, A327731.

nmax = 53; CoefficientList[Series[Product[1/(1 + (-x)^k)^DivisorSum[k, Mod[#, 2] &], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[(-1)^k Sum[(-1)^(k/d) d DivisorSum[d, Mod[#, 2] &], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

G.f.: Product_{k>=1} 1 / (1 + (-x)^k)^A001227(k).

A329467 Expansion of Product_{i>=1, j>=1} (1 + x^(ij)) (1 + x^(2ij)).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 31, 47, 81, 126, 204, 308, 487, 720, 1098, 1613, 2395, 3461, 5061, 7213, 10362, 14633, 20712, 28926, 40497, 56000, 77527, 106349, 145791, 198339, 269678, 364106, 491125, 658708, 882077, 1175392, 1563884, 2071363, 2739095, 3608040, 4744058, 6216087

Offset: 0

Ilya Gutkovskiy, Nov 13 2019

nonn

Weigh transform of A069735.

Cf. A000005, A001935, A006171, A069735, A107742, A320245.

nmax = 41; CoefficientList[Series[Product[((1 - x^(4 k))/(1 - x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d If[EvenQ[d], DivisorSigma[0, d] + DivisorSigma[0, d/2], DivisorSigma[0, d]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

G.f.: Product_{i>=1, j>=1} (1 + x^(2*i*j)) / (1 - x^(i*(2*j - 1))).
G.f.: Product_{k>=1} ((1 - x^(4*k)) / (1 - x^k))^A000005(k).
G.f.: Product_{k>=1} (1 + x^k)^A069735(k).

A329805 Expansion of Product_{i>=1, j>=1} (1 + x^(ij) + x^(2i*j)).

Original entry on oeis.org

1, 1, 3, 4, 10, 14, 27, 39, 71, 101, 170, 242, 390, 551, 852, 1198, 1803, 2512, 3687, 5101, 7351, 10088, 14289, 19472, 27208, 36810, 50790, 68262, 93200, 124453, 168279, 223364, 299473, 395214, 525754, 690103, 911644, 1190447, 1562449, 2030381, 2649162, 3426591

Offset: 0

Ilya Gutkovskiy, Nov 21 2019

nonn

Cf. A000005, A000726, A006171, A107742, A329467.

nmax = 41; CoefficientList[Series[Product[((1 - x^(3 k))/(1 - x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 - x^(3*k)) / (1 - x^k))^A000005(k).

cp's OEIS Frontend

A333653 Expansion of Product_{i>=1, j>=1} (1 + i*x^(i*j)).

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A356957 Number of set partitions of strict integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1.

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A318844 Expansion of Product_{k>=1} (1 + x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

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A327746 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i*(2*j - 1))).

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A329467 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j)) * (1 + x^(2*i*j)).

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A329805 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j) + x^(2*i*j)).

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A333653 Expansion of Product_{i>=1, j>=1} (1 + ix^(ij)).

A327746 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i(2j - 1))).

A329467 Expansion of Product_{i>=1, j>=1} (1 + x^(ij)) (1 + x^(2ij)).

A329805 Expansion of Product_{i>=1, j>=1} (1 + x^(ij) + x^(2i*j)).