A081362 -id:A081362 - cp's OEIS frontend

A286352 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)^k.

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 1, -1, 0, 1, -4, 3, -2, 1, 0, 1, -5, 6, -4, 4, -1, 0, 1, -6, 10, -8, 9, -4, 1, 0, 1, -7, 15, -15, 17, -12, 5, -1, 0, 1, -8, 21, -26, 30, -28, 15, -6, 2, 0, 1, -9, 28, -42, 51, -56, 38, -21, 9, -2, 0, 1, -10, 36, -64, 84

Offset: 0

Seiichi Manyama, May 08 2017

			Square array begins:
   1,  1,  1,  1,  1,   1, ...
   0, -1, -2, -3, -4,  -5, ...
   0,  0,  1,  3,  6,  10, ...
   0, -1, -2, -4, -8, -15, ...
   0,  1,  4,  9, 17,  30, ...

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

Columns k=0-32 give: A000007, A081362, A022597-A022627.
Main diagonal gives A255526.
Antidiagonal sums give A299208.
Cf. A286335.

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

A069911 Expansion of Product_{i in A069909} 1/(1 - x^i).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 20, 25, 29, 35, 41, 49, 57, 68, 78, 93, 107, 125, 144, 168, 192, 223, 255, 294, 335, 385, 437, 501, 568, 647, 732, 833, 939, 1065, 1199, 1355, 1523, 1717, 1925, 2166, 2425, 2720, 3040, 3405, 3797, 4244, 4727, 5272

Offset: 0

N. J. A. Sloane, May 05 2002

nonn

Arises from an identity of Slater's.
Number of partitions of 2*n+1 into distinct odd parts. - Vladeta Jovovic, May 08 2003
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of partitions of 2n+1 such that if k is the largest part, then k occurs an odd number of times and each integer from 1 to k-1 occurs a positive even number of times. Example: a(4)=2 because we have [3,2,2,1,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 16 2006
Difference between number of partitions of 2n+1 with an odd number of parts and those with an even number of parts (this is a consequence of Jovovic's comment above). - George Beck, May 22 2016
Let b(k) be the convolution inverse of A035457, k=1, 2, 3, ...; then a(n) = -b(4n+3), n = 0, 1, 2, 3, ... (conjectured). - George Beck, Aug 19 2017

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + ...
G.f. = q^23 + q^71 + q^119 + q^167 + 2*q^215 + 2*q^263 + 3*q^311 + 4*q^359 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities Quaestiones Math., 24 (2001), 403-416.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33--37. MR0541341 (80j:05010). See Theorem 3. [From _N. J. A. Sloane_, Mar 19 2012]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

Cf. A000700, A035457, A069908, A069909, A069910, A081362, A122129, A122135.

h:=product(1+x^(2*i-1),i=1..60): hser:=series(h,x=0,120): seq(coeff(hser,x^(2*n+1)),n=0..56); # Emeric Deutsch, Apr 16 2006

H[x_] := x*QPochhammer[-1/x, x^2]/(1 + x); s = (H[Sqrt[x]] - H[-Sqrt[x]]) / (2*Sqrt[x]) + O[x]^60; CoefficientList[s, x] (* Jean-François Alcover, Nov 14 2015, after Emeric Deutsch *)

{a(n) = my(A); if( n<0, 0, n=2*n+1; A = x * O(x^n); -polcoeff( eta(x + A) / eta(x^2 + A), n))}; /* Michael Somos, Jul 18 2006 */

a(n) = A000700(2n+1) = -A081362(2n+1).
Euler transform of period 16 sequence [ 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: ( H(sqrt(x)) - H(-sqrt(x)) ) / (2*sqrt(x)), where H(x)=prod(i>=1, 1+x^(2*i-1) ). - Emeric Deutsch, Apr 16 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 14 2015
Expansion of f(x, x^7) / f(-x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 04 2016

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

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

A181315 G.f. A(x) satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^n).

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 227, 832, 3125, 11970, 46579, 183614, 731688, 2942673, 11928707, 48688888, 199932987, 825379993, 3423614756, 14261439594, 59635806865, 250241613688, 1053380320889, 4446989542144, 18823433444211, 79871578901283

Offset: 0

Paul D. Hanna, Oct 16 2010

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 227*x^6 +...
The g.f. A = A(x) satisfies
log(A) = x*A/(1-x^2*A^2) + (x^2/2)*A^2/(1-x^4*A^4) + (x^3/3)*A^3/(1-x^6*A^6) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A081362, A109085, A270914, A366018, A366026.

nmax:=25: kmax:=nmax: for n from 1 to nmax+1 do A(x):=add(a(k)*x^k, k=0..kmax-1): A(x) := product((1 + x^k*A(x)^k),k=1..kmax+1): a(n-1):=coeff(A(x),x,n-1): od: seq(a(n),n=0..nmax); # Johannes W. Meijer, Jul 04 2011

InverseSeries[x QPochhammer[x, x^2] + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

{a(n)=polcoeff(1/x*serreverse(x/prod(k=1,n+1,1+x^k+x*O(x^n))),n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x*A+x*O(x^n))^m/(1-(x*A)^(2*m))/m))); polcoeff(A, n)}

G.f.: A(x) = Sum_{n>=0} A000009(n)*x^n*A(x)^n, where A000009(n) is the number of partitions of n into distinct parts.
G.f.: A(x) = (1/x)*Series_Reversion[x^(1/24)*eta(x)/eta(x^2)] (cf. A081362).
G.f. satisfies A(x) = exp( Sum_{n>=1} (x^n/n)*A(x)^n/(1 - (x*A(x))^(2*n)) ).
a(n) ~ c * d^n / n^(3/2), where d = A270914 = 4.50247674761735448773859393270078440676312875609162163346454... and c = A366018 = 0.482420439587319764659364391266849418507665645926542970519109122... - Vaclav Kotesovec, Aug 21 2018

A260876 Number of m-shape set partitions, square array read by ascending antidiagonals, A(m,n) for m, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 5, 5, 1, 1, 11, 31, 15, 7, 1, 1, 36, 365, 379, 52, 11, 1, 1, 127, 6271, 25323, 6556, 203, 15, 1, 1, 463, 129130, 3086331, 3068521, 150349, 877, 22, 1, 1, 1717, 2877421, 512251515, 3309362716, 583027547, 4373461, 4140, 30

Offset: 0

Peter Luschny, Aug 02 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.
If m = 0, all possible sizes are zero. Thus the number of set partitions of 0-shape is the number of integer partitions of n (partition numbers A000041).
If m = 1, the set is {1, 2, ..., n} and the set of all possible sizes are the integer partitions of n. Thus the number of set partitions of 1-shape is the number of set partitions (Bell numbers A000110).
If m = 2, the set is {1, 2, ..., 2n} and the number of set partitions of 2-shape is the number of set partitions into even blocks A005046.
From Petros Hadjicostas, Aug 06 2019: (Start)
Irwin (1916) proved the following combinatorial result: Assume r_1, r_2, ..., r_n are positive integers and we have r_1*r_2*...*r_n objects. We divide them into r_1 classes of r_2*r_3*...*r_n objects each, then each class into r_2 subclasses of r_3*...*r_n objects each, and so on. We call each such classification, without reference to order, a "classification" par excellence. He proved that the total number of classifications is (r_1*r_2*...*r_n)!/( r1! * (r_2!)^(r_1) * (r_3!)^(r_1*r_2) * ... (r_n!)^(r_1*r_2*...*r_{n-1}) ).
Apparently, this problem appeared in Carmichael's "Theory of Numbers".
This result can definitely be used to prove some special cases of my conjecture below. (End)

			[ n ] [0  1   2       3        4           5              6]
[ m ] ------------------------------------------------------
[ 0 ] [1, 1,  2,      3,       5,          7,            11]  A000041
[ 1 ] [1, 1,  2,      5,      15,         52,           203]  A000110
[ 2 ] [1, 1,  4,     31,     379,       6556,        150349]  A005046
[ 3 ] [1, 1, 11,    365,   25323,    3068521,     583027547]  A291973
[ 4 ] [1, 1, 36,   6271, 3086331, 3309362716, 6626013560301]  A291975
        A260878,A309725, ...
For example the number of set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] is 1, 84 and 280 respectively. Thus A(3,3) = 365.
Formatted as a triangle:
[1]
[1, 1]
[1, 1,   2]
[1, 1,   2,    3]
[1, 1,   4,    5,     5]
[1, 1,  11,   31,    15,    7]
[1, 1,  36,  365,   379,   52,  11]
[1, 1, 127, 6271, 25323, 6556, 203, 15]
.
From _Peter Luschny_, Aug 14 2019: (Start)
For example consider the case n = 4. There are five integer partitions of 4:
  P = [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. The shapes are m times the parts of the integer partitions: S(m) = [[4m], [3m, m], [2m, 2m], [2m, m, m], [m, m, m, m]].
* In the case m = 1 we look at set partitions of {1, 2, 3, 4} with sizes in  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] which gives rise to [1, 4, 3, 6, 1] with sum 15.
* In the case m = 2 we look at set partitions of {1, 2, .., 8} with sizes in [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]] which gives rise to [1, 28, 35, 210, 105] with sum 379.
* In the case m = 0 we look at set partitions of {} with sizes in [[0], [0, 0], [0, 0], [0, 0, 0], [0, 0, 0, 0]] which gives rise to [1, 1, 1, 1, 1] with sum 5 (because the only partition of the empty set is the set that contains the empty set, thus from the definition T(0,4) = Sum_{S(0)} card({0}) = A000041(4) = 5).
If n runs through 0, 1, 2,... then the result is an irregular triangle in which the n-th row lists multinomials for partitions of [m*n] which have only parts which are multiples of m. These are the triangles A080575 (m = 1), A257490 (m = 2), A327003 (m = 3), A327004 (m = 4). In the case m = 0 the triangle is A000012 subdivided into rows of length A000041. See the cross references how this case integrates into the full picture.
(End)

Alois P. Heinz, Antidiagonals n = 0..54, flattened
Frank Irwin, Solution to Problem 223 proposed by T. E. Mason in October 1914, Amer. Math. Monthly 23(9) (1916), 352-353.

-----------------------------------------------------------------
[m] |  multi-   |  sum of   |  main     |  by       |  comple-  |
    |  nomials  |  rows     |  diagonal |  size     |  mentary  |
-----------------------------------------------------------------
[0] |  A000012  |  A000041  |  A000012  |  A008284  |  A081362  |
[1] |  A036040  |  A000110  |  A000012  |  A048993  |  A000587  |
[2] |  A257490  |  A005046  |  A001147  |  A156289  |  A260884  |
[3] |  A327003  |  A291973  |  A025035  |  A291451  |  A291974  |
[4] |  A327004  |  A291975  |  A025036  |  A291452  |  A291976  |
Cf. A326996 (main diagonal), A260883 (ordered), A260875 (complementary).
Columns include A000012, A260878, A309725.

A:= proc(m, n) option remember; `if`(m=0, combinat[numbpart](n),
      `if`(n=0, 1, add(binomial(m*n-1, m*k-1)*A(m, n-k), k=1..n)))
    end:
seq(seq(A(d-n, n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 14 2019

A[m_, n_] := A[m, n] = If[m==0, PartitionsP[n], If[n==0, 1, Sum[Binomial[m n - 1, m k - 1] A[m, n - k], {k, 1, n}]]];
Table[Table[A[d - n, n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

def A260876(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum(SetPartitions(sum(s), s).cardinality() for s in shapes)
for m in (0..4): print([A260876(m,n) for n in (0..6)])

From Petros Hadjicostas, Aug 02 2019: (Start)
A(m, 2) = 1 + (1/2) * binomial(2*m, m) for m >= 1.
A(m, 3) = 1 + binomial(3*m, m) + (3*m)!/(6 * (m!)^3) for m >= 1.
A(m, 4) = (1/4!) * multinomial(4*m, [m, m, m, m]) + (1/2) * multinomial(4*m, [2*m, m, m]) + multinomial(4*m, [m, 3*m]) + (1/2) * multinomial(4*m, [2*m, 2*m]) + 1 for m >= 1.
Conjecture: For n >= 0, let P be the set of all possible lists (a_1,...,a_n) of nonnegative integers such that a_1*1 + a_2*2 + ... + a_n*n = n. Consider terms of the form multinomial(n*m, m*[1,..., 1,2,..., 2,..., n,..., n])/(a_1! * a_2! * ... * a_n!), where in the list [1,...,1,2,...,2,...,n,...,n] the number 1 occurs a_1 times, 2 occurs a_2 times, ..., and n occurs a_n times. (Here a_n = 0 or 1.) Summing these terms over P we get A(m, n) provided m >= 1. (End)
Conjecture for a recurrence: A(m, n) = Sum_{k = 0..n-1} binomial(m*n - 1, m*k) * A(m, k) with A(m, 0) = 1 for m >= 1 and n >= 0. (Unfortunately, the recurrence does not hold for m = 0.) - Petros Hadjicostas, Aug 12 2019

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).

			The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.

q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 12, 13, 17, 21, 25, 32, 39, 46, 58, 68, 83, 99, 121, 141, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

			The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A284314 Expansion of Product_{k>=0} (1 - x^(5*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 3, -2, 0, 0, -1, 3, -3, 1, 0, -1, 4, -4, 1, 0, -1, 4, -5, 2, 0, -1, 5, -7, 3, 0, -1, 5, -8, 5, -1, -1, 6, -10, 6, -1, -1, 6, -12, 9, -2, -1, 7, -14, 11, -3, -1, 7, -16, 15, -5

Offset: 0

Seiichi Manyama, Mar 25 2017

sign

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

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), A284313 (m=4), this sequence (m=5).

Cf. A280454, A284097.

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

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

a(n) = -(1/n)*Sum_{k=1..n} A284097(k)*a(n-k), a(0) = 1.

A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 2, 3, 2, 0, 2, 4, 4, 0, 1, 4, 6, 2, 1, 4, 8, 4, 2, 4, 10, 6, 2, 3, 12, 10, 4, 2, 13, 14, 8, 2, 14, 18, 12, 2, 14, 22, 18, 3, 14, 26, 26, 6, 14, 29, 34, 10, 14, 32, 44, 16, 14, 34, 56, 26, 16, 34, 67, 38, 20, 34, 78, 52, 26

Offset: 0

Gus Wiseman, Mar 08 2018

By Theorem 1 of Craig, the values a(n) in this list are known to be nonnegative. Combined with Theorem 2 of Seo and Yee, this shows that a(n) = |number of partitions of n into odd parts with an odd index minus the number of partitions of n into odd parts with an even index|. - William Craig, Dec 31 2021

Seunghyun Seo and Ae Ja Yee, Index of seaweed algebras and integer partitions, Electronic Journal of Combinatorics, 27:1 (2020), #P1.47. See Conjecture 1 and Theorem 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Gus Wiseman)
Vincent E. Coll, Andrew W. Mayers, and Nick W. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv:1809.09271 [math.CO], 2018.
William Craig, Seaweed Algebras and the Index Statistic for Partitions, arXiv:2112.09269 [math.CO], 2021.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A099323, A220418, A290261, A292043, A292137, A298118, A300575.

CoefficientList[Series[1/QPochhammer[x, -x^2], {x, 0, 100}], x]
nmax = 100; CoefficientList[Series[Product[1/((1+x^(4*k-1))*(1-x^(4*k-3))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 04 2019 *)

O.g.f.: Product_{n >= 0} 1/(1 - (-1)^n x^(2n+1)).
a(n) = Sum (-1)^k where the sum is over all integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.
a(n) has average order Gamma(1/4) * exp(sqrt(n/3)*Pi/2) / (2^(9/4) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jun 04 2019

A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 3, -2, 0, -1, 3, -3, 1, -1, 4, -4, 1, -1, 4, -5, 2, -1, 5, -7, 3, -1, 5, -8, 5, -2, 6, -10, 6, -2, 6, -12, 9, -3, 7, -14, 11, -4, 7, -16, 15, -6, 8, -19, 18, -8, 9, -21, 23, -11, 10, -24

Offset: 0

Seiichi Manyama, Mar 24 2017

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), this sequence (m=4), A284314 (m=5).

Cf. A050449, A169975, A284316.

V:= Vector(100):
V[1]:= 1:
for k from 0 to 24 do
  V[4*k+2..100]:= V[4*k+2..100] - V[1..99-4*k]
od:
convert(V,list); # Robert Israel, May 03 2017

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

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

a(n) = -(1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1.

O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n-1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284316. - Peter Bala, Nov 28 2020

cp's OEIS Frontend

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