A081362 -id:A081362 - cp's OEIS frontend

A264753 Irregular triangle read by rows: T(n,k) = A127671(n,k)/A036040(n,k), n >= 1 and 1 <= k <= A000041(n).

1, 1, -1, 1, -1, 2, 1, -1, -1, 2, -6, 1, -1, -1, 2, 2, -6, 24, 1, -1, -1, -1, 2, 2, 2, -6, -6, 24, -120, 1, -1, -1, -1, 2, 2, 2, 2, -6, -6, -6, 24, 24, -120, 720, 1, -1, -1, -1, -1, 2, 2, 2, 2, 2, -6, -6, -6, -6, -6, 24, 24, 24, -120, -120, 720, -5040

Offset: 1

Johannes W. Meijer, Jul 12 2016

This sequence connects the multinomial coefficients A036040 (M_3) with A127671 (M_5).
The numbers of terms in n-th row is the number of partitions A000041(n). The number of terms T(n, k) with equal values in the n-th row follow the rhythm of A008284(n).
Some row sums are [1, 0, 2, -5, 21, -104, 636, -4511, 36455, -330954, 3334390, -36914039].

			The first few rows of the T(n,k) triangle:
n = 1: 1
n = 2: 1, -1
n = 3: 1, -1, 2
n = 4: 1, -1, -1, 2, -6
n = 5: 1, -1, -1, 2, 2, -6, 24
n = 6: 1, -1, -1, -1, 2, 2, 2, -6, -6, 24, -120
n = 7: 1, -1, -1, -1, 2, 2, 2, 2, -6, -6, -6, 24, 24, -120, 720

Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.

Cf. A036040 (M_3), A127671 (M_5), A000041, A008284, A081362.

Cf. A048996 (M_0), A036038 (M_1), A036039 (M_2), A117506 (M_4).

nmax:=8: with(combinat): A008284 := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else A008284(n-1, k-1) + A008284(n-k, k) fi end: for n from 1 to nmax do p:=0: k:=1: while k < numbpart(n)+1 do p := p+1: k1 := A008284(n, p): while k1 > 0 do A264753(n, k) := (-1)^(p+1)*(p-1)!: k := k+1: k1 := k1-1: od: od: od: seq(seq(A264753(n, k), k = 1..numbpart(n)), n = 1..nmax);

nMax = 8; A008284[n_, k_] := A008284[n, k] = If[k<0 || n<0, 0, If[k == 0, If[n == 0, 1, 0], A008284[n-1, k-1] + A008284[n-k, k]]]; For[n = 1, n <= nMax, n++, p = 0; k = 1; While[k < PartitionsP[n]+1, p = p+1; k1 = A008284[n, p]; While[k1>0, A264753[n, k] = (-1)^(p+1)*(p-1)!; k = k+1; k1 = k1-1]]]; Table[Table[A264753[n, k], {k, 1, PartitionsP[n]}], {n, 1, nMax}] // Flatten (* Jean-François Alcover, Oct 01 2016, translated from Maple *)

T(n, k) = A127671(n, k)/A036040(n, k), n >= 1 and 1 <= k <= A000041(n).

A284321 Expansion of Product_{k>=0} (1 - x^(5k+1))(1 - x^(5*k+4)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 1, 0, -1, 2, -2, 1, 1, -2, 3, -3, 2, 0, -3, 5, -5, 3, 1, -5, 7, -7, 4, 1, -7, 11, -11, 6, 2, -10, 15, -15, 9, 2, -14, 22, -22, 12, 4, -20, 30, -29, 17, 4, -27, 42, -41, 23, 7, -37, 55, -54, 31, 8, -49, 76, -74, 41, 12, -66, 99, -96, 55, 14

Offset: 0

Seiichi Manyama, Mar 25 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A003114, A284150, A284322, A284314, A284317.

Cf. Product_{k>=0} (1 - x^(m*k+1))*(1 - x^(m*k+m-1)): A137569 (m=3), A081362 (m=4), this sequence (m=5), A109389 (m=6).

CoefficientList[Series[Product[(1 - x^(5k + 1)) ( 1 - x^(5k + 4)), {k, 0, 100}], {x, 0, 100}],x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, (1 - x^(5*k + 1)) * (1 - x^(5*k + 4))) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1.

A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 1, -1, 1, -1, -15, -20, 0, 0, 1, 1, -1, -31, -66, -8, 11, 4, -1, 1, -1, -63, -212, -54, 99, 42, 2, 2, 1, -1, -127, -666, -284, 725, 455, 63, 8, -2, 1, -1, -255, -2060, -1350, 4935, 4580, 958, 73

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
   0, -1, -3,  -7, -15,  -31, ...
  -1, -2, -6, -20, -66, -212, ...
   1,  1,  0,  -8, -54, -284, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give A081362, A255528, A284896, A284897, A284898, A284899.

Cf. A144048, A284992.

G.f. of column k: Product_{j>=1} 1/(1+x^j)^(j^k).

A292560 Expansion of Product_{k>=1} 1/(1 + x^(k^3)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 19 2017

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Convolution inverse of A279329.
The difference between the number of partitions of n into an even number of cubes and the number of partitions of n into an odd number of cubes.
In general, if m > 0 and g.f. = Product_{k>=1} 1/(1 + x^(k^m)), then a(n) ~ (-1)^n * exp((m+1) * (Gamma(1/m) * Zeta(1 + 1/m) / m^2)^(m/(m+1)) * n^(1/(m+1)) / 2) * (Gamma(1/m) * Zeta(1 + 1/m))^(m/(2*(m+1))) / (sqrt(Pi*(m+1)) * 2^((m+1)/2) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))). - Vaclav Kotesovec, Sep 19 2017

Cf. A081362 (m=1), A292520 (m=2).

Cf. A003108, A279329, A279484.

nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]

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

a(n) ~ (-1)^n * exp(2 * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(3/8) / (8 * 3^(1/4) * sqrt(Pi) * n^(7/8)). - Vaclav Kotesovec, Sep 19 2017

A293285 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(Sum_{j=0..k} j!x^(ji)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -2, -1, 0, 1, -1, -2, 3, 1, 0, 1, -1, -2, -3, -1, -1, 0, 1, -1, -2, -3, 11, -5, 1, 0, 1, -1, -2, -3, -13, 7, 9, -1, 0, 1, -1, -2, -3, -13, 55, -15, 3, 2, 0, 1, -1, -2, -3, -13, -65, 33, -63, -20, -2, 0, 1, -1, -2, -3, -13, -65

Offset: 0

Seiichi Manyama, Oct 04 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   0, -1, -1, -1,  -1, ...
   0,  0, -2, -2,  -2, ...
   0, -1,  3, -3,  -3, ...
   0,  1, -1, 11, -13, ...
   0, -1, -5,  7,  55, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000007, A081362, A293287.
Rows n=0..1 give A000012, (-1)*A057427.
Main diagonal gives A293251.
Cf. A293202, A293293.

A293300 E.g.f.: Product_{m>0} 1/(1 + x^m).

Original entry on oeis.org

1, -1, 0, -6, 24, -120, 720, -5040, 80640, -725760, 7257600, -79833600, 1437004800, -18681062400, 261534873600, -5230697472000, 104613949440000, -1778437140480000, 32011868528640000, -729870602452992000, 17030314057236480000, -408727537373675520000

Offset: 0

Seiichi Manyama, Oct 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..446

Column k=1 of A293299.

Cf. A081362.

nn = 30; x = 'x + O('x^nn); Vec(serlaplace(prod(m=1, nn, 1/(1 + x^m )))) \\ Michel Marcus, Oct 05 2017

a(n) = n! * A081362(n).

A294460 E.g.f.: exp(-Sum_{n>=1} A000593(n) * x^n).

Original entry on oeis.org

1, -1, -1, -19, 73, -401, 5191, -29779, 879089, -7232833, 103048111, -1891058291, 31696845049, -649348332049, 9310670445623, -270217657103731, 5480877008565601, -131578355696804609, 3133521575795986399, -81890613282163881043, 2460096066325021029161

Offset: 0

Seiichi Manyama, Oct 31 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..437

E.g.f.: exp(-Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294459 (k=0), this sequence (k=1), A294461 (k=2).

Cf. A000593, A081362, A294394.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d*(d%2))*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000593(k)*a(n-k)/(n-k)! for n > 0.

A300548 a(n) = [x^n] Product_{d|n} 1/(1 + x^d).

Original entry on oeis.org

1, -1, 0, -2, 0, -2, 0, -2, 0, -5, 1, -2, 1, -2, 0, -14, 0, -2, 1, -2, 0, -18, 0, -2, 0, -7, 1, -23, 0, -2, 6, -2, 0, -26, 1, -26, 4, -2, 0, -30, 0, -2, 6, -2, 1, -286, 0, -2, 0, -9, 7, -38, 0, -2, 8, -38, 1, -42, 1, -2, 7, -2, 0, -493, 0, -44, 9, -2, 0, -50, 10, -2, 0, -2, 1, -698, 1, -50, 12, -2, 0, -239, 1, -2, 10, -56

Offset: 0

Ilya Gutkovskiy, Mar 08 2018

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Antti Karttunen, Table of n, a(n) for n = 0..10201
Index entries for sequences related to partitions

Cf. A018818, A033630, A065091, A081362, A300547, A300549.

Table[SeriesCoefficient[Product[1/(1 + Boole[Mod[n, k] == 0] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 85}]

A300548(n) = if(!n, 1, my(p=1); fordiv(n, d, p /= (1 + 'x^d)); polcoeff(Ser(p, 'x, 1+n), n)); \\ Antti Karttunen, Nov 27 2024

a(n) = -2 if n is an odd prime (A065091).

A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

Original entry on oeis.org

1, 1, 0, -1, -1, 1, 1, -1, -2, 0, 2, 0, -3, -1, 3, 2, -3, -3, 3, 4, -3, -6, 2, 7, -1, -8, 0, 10, 2, -11, -4, 12, 7, -13, -10, 13, 13, -13, -17, 13, 22, -11, -26, 9, 31, -6, -36, 2, 41, 3, -46, -9, 51, 17, -55, -26, 59, 36, -62, -48, 63, 61, -64, -75, 64, 92, -60, -109, 55, 127, -48, -147, 37, 167

Offset: 0

Gus Wiseman, Mar 09 2018

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

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A220418, A290261, A292043, A292137, A300574.

CoefficientList[Series[QPochhammer[-x,-x^2],{x,0,100}],x]

O.g.f.: Product_{n >= 0} (1 + (-1)^n x^(2n+1)).

a(n) = Sum (-1)^k where the sum is over all strict integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.

A302236 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 + x^k).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, -1, 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 1, 0, 0, 0, -1, 1, 0, -1, 1, -2, 1, 0, 0, 2, -1, 0, 0, -1, 2, -1, -1, 1, -2, 1, 0, 0, 0, -2, -1, 2, 0, 0, 1, -3, 2, -1, 1, 2, -2, -1, -1, 1, 3, 0, -2, 1, -2, 0, 3, 0, 0, -2, -2, 5, 1, 1, -1, -4, 1, -1, 2, 4, -2

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

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The difference between the number of partitions of n into an even number of nonprime parts and the number of partitions of n into an odd number of nonprime parts.

Convolution of the sequences A000586 and A081362.

Index entries for sequences related to partitions

Cf. A000586, A002095, A018252, A048165, A081362, A096258, A302234.

nmax = 80; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[1/(1 + Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 + x^A018252(k)).

cp's OEIS Frontend

A264753 Irregular triangle read by rows: T(n,k) = A127671(n,k)/A036040(n,k), n >= 1 and 1 <= k <= A000041(n).

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A284321 Expansion of Product_{k>=0} (1 - x^(5*k+1))*(1 - x^(5*k+4)) in powers of x.

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A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

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A292560 Expansion of Product_{k>=1} 1/(1 + x^(k^3)).

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A293285 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(Sum_{j=0..k} j!*x^(j*i)).

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A293300 E.g.f.: Product_{m>0} 1/(1 + x^m).

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A294460 E.g.f.: exp(-Sum_{n>=1} A000593(n) * x^n).

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A300548 a(n) = [x^n] Product_{d|n} 1/(1 + x^d).

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A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

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A284321 Expansion of Product_{k>=0} (1 - x^(5k+1))(1 - x^(5*k+4)) in powers of x.

A293285 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(Sum_{j=0..k} j!x^(ji)).