A261568 -id:A261568 - cp's OEIS frontend

A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

1, 2, 2, 6, 6, 10, 18, 22, 30, 42, 66, 78, 110, 138, 186, 254, 318, 402, 522, 654, 822, 1074, 1306, 1638, 2022, 2514, 3058, 3798, 4662, 5658, 6882, 8358, 10062, 12186, 14610, 17534, 21150, 25146, 29994, 35694, 42446, 50178, 59514, 70110, 82758, 97602, 114570, 134262

Offset: 0

Christian G. Bower, Apr 01 1998

nonn

"EFK" (unordered, size, unlabeled) transform of 2,2,2,2,...
Number of partitions into distinct parts of 2 sorts, see example. - Joerg Arndt, May 22 2013
In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016
Antidiagonal sums of A284593. - Peter Bala, Mar 30 2017

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(7) = 22 partitions of 7 into distinct parts of 2 sorts (format P:S for part:sort):
01:  [ 1:0  2:0  4:0  ]
02:  [ 1:0  2:0  4:1  ]
03:  [ 1:0  2:1  4:0  ]
04:  [ 1:0  2:1  4:1  ]
05:  [ 1:0  6:0  ]
06:  [ 1:0  6:1  ]
07:  [ 1:1  2:0  4:0  ]
08:  [ 1:1  2:0  4:1  ]
09:  [ 1:1  2:1  4:0  ]
10:  [ 1:1  2:1  4:1  ]
11:  [ 1:1  6:0  ]
12:  [ 1:1  6:1  ]
13:  [ 2:0  5:0  ]
14:  [ 2:0  5:1  ]
15:  [ 2:1  5:0  ]
16:  [ 2:1  5:1  ]
17:  [ 3:0  4:0  ]
18:  [ 3:0  4:1  ]
19:  [ 3:1  4:0  ]
20:  [ 3:1  4:1  ]
21:  [ 7:0  ]
22:  [ 7:1  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Moussa Ahmia and Mohammed L. Nadji, Overpartitions into distinct parts, Comb. Number Theory 14 (2025), no. 1, 65-74, preprint (2021).
C. G. Bower, Transforms (2)
Vaclav Kotesovec, Asymptotic formula for A032302
Eric Weisstein's World of Mathematics, Dilogarithm
Eric Weisstein's MathWorld, Polylogarithm
Wikipedia, Polylogarithm

Cf. A000009, A032308, A261562, A261568, A261569, A266576, A284593.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-2,x,99)/3,x)):
seq(coeff(%,x,n), n=0..47); # Peter Luschny, Nov 17 2016

nn=47; CoefficientList[Series[Product[1+2x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 07 2013 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-2, x]/3 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+2*x^n)) \\ Joerg Arndt, May 22 2013

a(n) = A072706(n)*2 for n>=1.
G.f.: Sum_{n>=0} (2^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k ) ). - Joerg Arndt, Jan 20 2014
a(n) = (1/3) [x^n] QPochhammer(-2,x). - Vladimir Reshetnikov, Nov 20 2015
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/6 + log(2)^2/2 + polylog(2, -1/2) = 1.43674636688368094636290202389358335424... . Equivalently, c = A266576 = Pi^2/12 + log(2)^2 + polylog(2, 1/4)/2. - Vaclav Kotesovec, Jan 04 2016

A032308 Expansion of Product_{k>=1} (1 + 3*x^k).

Original entry on oeis.org

1, 3, 3, 12, 12, 21, 48, 57, 84, 120, 228, 264, 399, 516, 732, 1119, 1416, 1884, 2532, 3324, 4296, 6168, 7545, 9984, 12684, 16500, 20577, 26688, 34572, 43032, 54264, 68232, 84972, 106176, 131664, 162507, 205680, 249888, 308856, 377796, 465195, 564024, 691788, 835572, 1017768, 1241040

Offset: 0

Christian G. Bower

nonn

"EFK" (unordered, size, unlabeled) transform of 3,3,3,3,...

Number of partitions into distinct parts of 3 sorts, see example. [Joerg Arndt, May 22 2013]

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(5) = 21 partitions of 5 into distinct parts of 3 sorts (format P:S for part:sort):
01:  [ 1:0  4:0  ]
02:  [ 1:0  4:1  ]
03:  [ 1:0  4:2  ]
04:  [ 1:1  4:0  ]
05:  [ 1:1  4:1  ]
06:  [ 1:1  4:2  ]
07:  [ 1:2  4:0  ]
08:  [ 1:2  4:1  ]
09:  [ 1:2  4:2  ]
10:  [ 2:0  3:0  ]
11:  [ 2:0  3:1  ]
12:  [ 2:0  3:2  ]
13:  [ 2:1  3:0  ]
14:  [ 2:1  3:1  ]
15:  [ 2:1  3:2  ]
16:  [ 2:2  3:0  ]
17:  [ 2:2  3:1  ]
18:  [ 2:2  3:2  ]
19:  [ 5:0  ]
20:  [ 5:1  ]
21:  [ 5:2  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
C. G. Bower, Transforms (2)

Cf. A000009, A032302, A261568, A261569.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 3*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-3,x,99),x)/4):
seq(coeff(%,x,n), n=0..45); # Peter Luschny, Nov 17 2016

nmax = 40; CoefficientList[Series[Product[1 + 3*x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-3, x]/4 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+3*x^n)) \\ Joerg Arndt, May 22 2013

G.f.: Product_{k>=1} (1 + 3*x^k).
a(n) = (1/4) * [x^n] QPochammer(-3, x). - Vladimir Reshetnikov, Nov 20 2015
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/6 + log(3)^2/2 + polylog(2, -1/3) = 1.93937542076670895307727171917789144122... . - Vaclav Kotesovec, Jan 04 2016
G.f.: Sum_{i>=0} 3^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

a(0) prepended and more terms added by Joerg Arndt, May 22 2013

A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

Original entry on oeis.org

1, 5, 5, 30, 30, 55, 180, 205, 330, 480, 1230, 1380, 2255, 3030, 4530, 8555, 10680, 15330, 21330, 29730, 39480, 67380, 81505, 116280, 153030, 210930, 270805, 370080, 534330, 675480, 900480, 1180380, 1544130, 1997280, 2597280, 3304805, 4581180, 5653080

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A032302, A032308, A261568, A291698.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 5*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015

nmax = 40; CoefficientList[Series[Product[1 + 5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*5^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-5, x]/6 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/6 + log(5)^2/2 + polylog(2, -1/5) = 2.74927912606080829002558751537626864449... . - Vaclav Kotesovec, Jan 04 2016

G.f.: Sum_{i>=0} 5^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A303350 Expansion of Product_{n>=1} (1 + 4*x^n)^(1/2).

Original entry on oeis.org

1, 2, 0, 10, -10, 38, -76, 310, -960, 3190, -10672, 37262, -130170, 459690, -1639940, 5901498, -21376154, 77900710, -285457200, 1051118590, -3887169486, 14431323506, -53766825940, 200964040290, -753348868380, 2831669141514, -10670007388128

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = -4.

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

Expansion of Product_{n>=1} (1 + b^2*x^n)^(1/b): A000009 (b=1), this sequence (b=2), A303351 (b=3).

Cf. A261568, A303347, A303352.

seq(coeff(series(mul((1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[(1 + 4*x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+4*x^k)^(1/2)))

a(n) ~ -(-1)^n * sqrt(c) * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=2} (1 + 4*(-1/4)^k) = 1.1864623436704848646891654544376222586... - Vaclav Kotesovec, Apr 22 2018

A344063 Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

Original entry on oeis.org

1, 1, 4, 20, 80, 384, 1600, 7424, 30720, 143360, 593920, 2703360, 11403264, 51118080, 214958080, 965738496, 4047503360, 17951621120, 76168560640, 334202142720, 1411970498560, 6211596451840, 26203595472896, 114246130073600, 484815908372480, 2101441598586880, 8896148580335616

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A261568, A304961, A338673, A340103, A344062, A344064, A344065, A344066, A344067, A344068.

nmax = 26; CoefficientList[Series[Product[(1 + 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 4^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 26}]

seq(n)={Vec(prod(k=1, n, 1 + 4^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 4^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/4))^(1/4) * 4^n * exp(2*sqrt(-polylog(2, -1/4)*n)) / (2*sqrt(5*Pi/4)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A286957 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 2, 0, 1, 4, 3, 6, 2, 0, 1, 5, 4, 12, 6, 3, 0, 1, 6, 5, 20, 12, 10, 4, 0, 1, 7, 6, 30, 20, 21, 18, 5, 0, 1, 8, 7, 42, 30, 36, 48, 22, 6, 0, 1, 9, 8, 56, 42, 55, 100, 57, 30, 8, 0, 1, 10, 9, 72, 56, 78, 180, 116, 84, 42, 10, 0, 1, 11, 10, 90, 72, 105, 294, 205, 180, 120, 66, 12, 0

Offset: 0

Ilya Gutkovskiy, May 17 2017

A(n,k) is the number of partitions of n into distinct parts of k sorts: the parts are unordered, but not the sorts.

			Square array begins:
1,  1,   1,   1,   1,   1,  ...
0,  1,   2,   3,   4,   5,  ...
0,  1,   2,   3,   4,   5,  ...
0,  2,   6,  12,  20,  30,  ...
0,  2,   6,  12,  20,  30,  ...
0,  3,  10,  21,  36,  55,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-5 give: A000007, A000009, A032302, A032308, A261568, A261569.
Main diagonal gives A291698.
Cf. A246935.

Table[Function[k, SeriesCoefficient[Product[(1 + k x^i), {i, 1, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-k, x]/(1 + k), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

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

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

Original entry on oeis.org

1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..1600

Cf. A261568, A246936, A067855, A303350.

Cf. A015128, A261584, A303390.

N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019

nmax = 25; CoefficientList[Series[Product[(1+4*x^k)/(1-4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = QPochhammer[-1, 1/4] / QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

A343466 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-4)^d.

Original entry on oeis.org

4, -6, 24, -66, 208, -676, 2344, -8226, 29144, -104760, 381304, -1398476, 5162224, -19172796, 71582944, -268439586, 1010580544, -3817734596, 14467258264, -54975633768, 209430787824, -799644629556, 3059510616424, -11728124734476, 45035996273872, -173215367702376, 667199944815064

Offset: 1

Ilya Gutkovskiy, Apr 16 2021

sign

Cf. A000010, A001868, A038065, A074763, A261568, A343465, A343467.

Table[-(1/n) Sum[EulerPhi[n/d] (-4)^d, {d, Divisors[n]}], {n, 1, 27}]
nmax = 27; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 4 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} phi(k) * log(1 + 4*x^k) / k.
a(n) = -(1/n) * Sum_{k=1..n} (-4)^gcd(n,k).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A261568.

A303392 Expansion of Product_{k>=1} ((1 + 4x^k) / (1 - 4x^k))^(1/2).

Original entry on oeis.org

1, 4, 12, 52, 156, 612, 2028, 7892, 27324, 107396, 384844, 1520436, 5566876, 22069796, 81990252, 325707348, 1222582268, 4862950020, 18395472460, 73233825524, 278700724764, 1110232691108, 4245596648876, 16920914168148, 64963831455996, 259012955299396

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Cf. A261568, A246936, A067855, A303350, A303391, A303360.

nmax = 30; CoefficientList[Series[Product[((1+4*x^k)/(1-4*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/2), {x, 0, nmax}], x]

a(n) ~ sqrt(c) * 4^n / sqrt(Pi*n), where c = QPochhammer[-1, 1/4]/QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

cp's OEIS Frontend

A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A032308 Expansion of Product_{k>=1} (1 + 3*x^k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A303350 Expansion of Product_{n>=1} (1 + 4*x^n)^(1/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A344063 Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A286957 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A303391 Expansion of Product_{k>=1} (1 + 4*x^k)/(1 - 4*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

A303392 Expansion of Product_{k>=1} ((1 + 4x^k) / (1 - 4x^k))^(1/2).