A081362 -id:A081362 - cp's OEIS frontend

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

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Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 2, -2, 1, 0, -1, 0, 2, -3, 3, -1, -1, 1, 1, -4, 5, -3, 0, 2, 0, -4, 7, -6, 1, 3, -2, -3, 9, -10, 4, 3, -5, -1, 11, -15, 10, 1, -8, 3, 10, -20, 17, -3, -10, 9, 7, -24, 26, -10, -10, 15, 2, -27, 37, -21, -8, 22, -6, -28, 49, -36, -2, 30, -19, -24, 61, -56, 10, 35

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

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Cf. A035382, A081360, A081362, A109389, A132462, A261612, A262928, A284312, A374064.

nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

a(0) = 1; a(n) = -Sum_{k=1..n} A261612(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132462(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A262928(n-k).

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

N. J. A. Sloane

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

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).

Column k=5 of A286352.

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

a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A260875 Square array read by ascending antidiagonals: number of m-shape complementary Bell numbers.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, 0, -1, 1, -1, 2, 1, 1, 1, -1, 9, -1, 1, -1, 1, -1, 34, -197, -43, -2, 1, 1, -1, 125, -5281, 6841, 254, -9, -1, 1, -1, 461, -123124, 2185429, -254801, 4157, -9, 2, 1, -1, 1715, -2840293, 465693001, -1854147586, -3000807, -70981, 50, -2

Offset: 1

Peter Luschny, Aug 09 2015

A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n.
M-complementary Bell numbers count the m-shape set partitions which have even length minus the number of such partitions which have odd length.
If m=0 all possible sizes are zero. Thus in this case the complementary Bell numbers count the integer partitions of n into an even number of parts minus the number of integer partitions of n into an odd number of parts (A081362).
If m=1 the set is {1,2,...,n} and the complementary Bell numbers count the set partitions which have even length minus the set partitions which have odd length (A000587).
If m=2 the set is {1,2,...,2n} and the complementary Bell numbers count the set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A260884).

			[ n ] [ 0   1   2      3        4            5              6]
[ m ] --------------------------------------------------------
[ 0 ] [ 1, -1,  0,    -1,       1,          -1,             1] A081362
[ 1 ] [ 1, -1,  0,     1,       1,          -2,            -9] A000587
[ 2 ] [ 1, -1,  2,    -1,     -43,         254,          4157] A260884
[ 3 ] [ 1, -1,  9,  -197,    6841,     -254801,      -3000807]
[ 4 ] [ 1, -1, 34, -5281, 2185429, -1854147586, 2755045819549]
      A010763,
For example the number of set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 84, 280 respectively. Thus A(3,3) = -1 + 84 - 280 = -197.
Formatted as a triangle:
[1]
[1, -1]
[1, -1,   0]
[1, -1,   0,    -1]
[1, -1,   2,     1,    1]
[1, -1,   9,    -1,    1,  -1]
[1, -1,  34,  -197,  -43,  -2,  1]
[1, -1, 125, -5281, 6841, 254, -9, -1]

Cf. A000587, A010763, A081362, A260877, A260833, A260884, A260876.

def A260875(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum((-1)^len(s)*SetPartitions(sum(s),s).cardinality() for s in shapes)
for m in (0..4): print([A260875(m,n) for n in (0..6)])

A303130 Expansion of Product_{n>=1} (1 + (9*x)^n)^(-1/3).

Original entry on oeis.org

1, -3, -9, -288, 459, -19278, -1539, -1265301, 10734525, -147277926, 520204923, -7511358663, 88687160577, -668191863951, 5357547144702, -87542760890124, 967961569696722, -5115624735401361, 46065749188891275, -430898393089547667, 6203508335817169257

Offset: 0

Seiichi Manyama, Apr 19 2018

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This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = -9^n.

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

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), this sequence (b=3), A303131 (b=4), A303132 (b=5).

Cf. A303074.

CoefficientList[Series[(2/QPochhammer[-1, 9*x])^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, N, (1 + (9*x)^k)^(-1/3))) \\ Altug Alkan, Apr 20 2018

a(n) ~ (-1)^n * exp(Pi*sqrt(n/18)) * 3^(2*n - 1/2) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A330373 Sum of all parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 60, 80, 85, 90, 114, 140, 168, 176, 207, 264, 300, 312, 378, 448, 493, 540, 620, 736, 825, 884, 1015, 1188, 1295, 1406, 1599, 1840, 2009, 2184, 2451, 2772, 3060, 3312, 3666, 4176, 4557, 4900, 5457, 6084, 6625, 7182, 7920, 8792, 9576, 10324, 11328, 12540

Offset: 0

Omar E. Pol, Dec 17 2019

nonn

a(n) is the sum of all parts of all partitions of n whose Ferrers diagrams are symmetric.
The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.
The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore, a(n) is also the sum of all parts of all partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

			For n = 10 there are only two partitions of 10 whose Ferrers diagram are symmetric, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1] as shown below:
  * * * * *
  * *
  *
  *
  *
            * * * *
            * * *
            * *
            *
The sum of all parts of these partitions is 5 + 2 + 1 + 1 + 1 + 4 + 3 + 2 + 1 = 20, so a(10) = 20.
Also, in accordance with the first formula; a(10) = 2*10 = 20.

Freddy Barrera, Table of n, a(n) for n = 0..10000

Row sums of A330372.
For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.
Cf. A000700, A066186, A081362, A235324.

seq(n)={Vec(deriv(exp(sum(k=1, n, x^k/(k*(1 - (-x)^k)) + O(x*x^n)))), -(n+1))} \\ Andrew Howroyd, Dec 31 2019

a(n) = n*A000700(n).
a(n) = abs(n*A081362(n)).
a(n) = abs(A235324(n)), n >= 1.

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

Original entry on oeis.org

1, -1, 0, -1, 0, 1, -1, 1, 0, 0, 1, 0, -1, 1, -1, 1, 0, -1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, 1, -1, 0, 0, -1, 2, 0, -1, 1, 0, -1, 2, -2, 0, 1, -1, 0, 1, -1, 0, 1, -2, 1, 1, -2, 1, 0, -2, 2, 0, -2, 2, -1, 0, 2, -1, -1, 1, -1, -1, 3, -2, 0, 2, -2, 1, 2, -3, 1, 1, -2, 2, 1

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

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Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035360, A035386, A081362, A082051, A132463, A137569, A274719, A284312, A374060.

nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 2)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082051(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} A035360(k) * a(n-k).
a(n) = Sum_{k=0..n} A010815(k) * A035386(n-k).

A374060 Expansion of Product_{k>=1} (1 - x^(3k-1)) (1 - x^(3*k)).

Original entry on oeis.org

1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 0, 1, 0, 0, 1, 0, -1, 0, 0, -1, 0, 1, -1, -1, 0, -1, 0, 1, 0, -1, 1, 0, -1, 1, 1, -1, 0, 1, 0, 1, 1, -1, 0, 1, -1, -1, 2, 0, -1, 1, 0, -1, 1, 0, -2, 0, 0, -1, 1, 1, -2, 0, 1, -2, 0, 2, -1, -1, 1, -1, -1, 2, -1, -1, 2, 0, -1, 2, 1, -2, 1, 0, -2, 2, 1, -2

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

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Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035361, A035382, A081362, A082050, A132462, A137569, A274719, A284315, A374058.

nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 1)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082050(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} A035361(k) * a(n-k).
a(n) = Sum_{k=0..n} A010815(k) * A035382(n-k).

A246582 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, -1, 1, -2, 1, -3, 1, -4, 2, -5, 3, -6, 5, -7, 7, -8, 10, -10, 13, -12, 17, -15, 21, -19, 26, -24, 31, -30, 38, -38, 45, -47, 54, -58, 64, -71, 77, -86, 91, -103, 109, -124, 129, -147, 154, -174, 182, -205, 216, -241, 254, -282, 300, -330, 351, -384, 412, -447, 480, -519, 560, -602, 649, -696, 753, -805

Offset: 0

N. J. A. Sloane, Aug 31 2014

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Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=3.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

For k=0 and 1 we get A081362, A027349 (apart from signs). Cf. A246581, A246583.

fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(3);

nmax = 100; CoefficientList[Series[x^6/((1-x)*(1-x^2)*(1-x^3)) * Product[1/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2016 *)

a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(n/6)*Pi) / (2^(13/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 12 2016

A246583 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, -1, 2, -2, 3, -4, 3, -6, 5, -9, 6, -12, 10, -16, 13, -20, 20, -26, 26, -32, 37, -41, 47, -51, 63, -65, 78, -81, 101, -103, 123, -128, 155, -161, 187, -199, 232, -247, 278, -302, 341, -371, 407, -449, 495, -545, 589, -654, 711, -786, 843, -936, 1011, -1116, 1194, -1320, 1423, -1563, 1674

Offset: 0

N. J. A. Sloane, Aug 31 2014

sign

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

For k=0 and 1 we get A081362, A027349 (apart from signs). Cf. A246581, A246582.

fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(4);

nmax = 100; CoefficientList[Series[x^10/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) * Product[1/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2016 *)

a(n) ~ (-1)^n * 3^(3/4) * n^(1/4) * exp(sqrt(n/6)*Pi) / (2^(15/4)*Pi^2). - Vaclav Kotesovec, Mar 12 2016

cp's OEIS Frontend

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

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A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

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A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

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A260875 Square array read by ascending antidiagonals: number of m-shape complementary Bell numbers.

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A303130 Expansion of Product_{n>=1} (1 + (9*x)^n)^(-1/3).

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A330373 Sum of all parts of all self-conjugate partitions of n.

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A374058 Expansion of Product_{k>=1} (1 - x^(3*k-2)) * (1 - x^(3*k)).

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A374060 Expansion of Product_{k>=1} (1 - x^(3*k-1)) * (1 - x^(3*k)).

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A246582 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 3.

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A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

A374060 Expansion of Product_{k>=1} (1 - x^(3k-1)) (1 - x^(3*k)).