A261734 -id:A261734 - cp's OEIS frontend

A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.

0, 1, 1, 2, 1, 4, 4, 7, 6, 13, 13, 21, 21, 36, 38, 57, 59, 90, 98, 137, 148, 210, 231, 310, 341, 459, 511, 664, 737, 957, 1073, 1357, 1518, 1918, 2156, 2673, 3002, 3712, 4182, 5100, 5737, 6976, 7866, 9460, 10652, 12777, 14402, 17126, 19284, 22867, 25761, 30340, 34139, 40099

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(6) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].

Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.
Index entries for sequences related to partitions

Cf. A001935, A027193, A042968, A261734, A339404, A339405, A339406.

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

nmax = 53; CoefficientList[Series[(1/2) (Product[(1 - x^(4 k))/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^(4 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A001935(n) - A261734(n)) / 2.

A134747 Expansion of q * (chi(-q) / chi(-q^4))^8 in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -8, 28, -64, 142, -352, 792, -1536, 2917, -5744, 10868, -19200, 33414, -58816, 101256, -167936, 275314, -452392, 732748, -1160064, 1819808, -2851104, 4421064, -6752256, 10236407, -15476272, 23215192, -34450944, 50811638, -74701632, 109138272, -158171136

Offset: 1

Michael Somos, Nov 07 2007

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 142*q^5 - 352*q^6 + 792*q^7 - 1536*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A131123, A261734.

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^2] QPochhammer[ -q^4, q^4])^8, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( (eta(x + A) * eta(x^8 + A)) / (eta(x^2 + A) * eta(x^4 + A)) )^8, n))};

Expansion of k * (1 - k) / ( 4 * (1 + k) ) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
Expansion of ( (eta(q) * eta(q^8)) / (eta(q^2) * eta(q^4)) )^8 in powers of q.
Euler transform of period 8 sequence [ -8, 0, -8, 8, -8, 0, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 16 * u*w * (v*w-1) * (v*u-1) - (v - u^2) * (v - w^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t).
G.f.: x * ( Product_{k>0} (1 + x^(4*k)) / (1 + x^k) )^8.
Convolution inverse of A131123.
Convolution 8th power of A261734. - Michael Somos, Oct 16 2015
a(n) ~ -(-1)^n * exp(Pi*sqrt(2*n)) / (2^(21/4) * n^(3/4)). - Vaclav Kotesovec, Apr 10 2018
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = -8 - 6*sqrt(2) + (3/2)*sqrt(60 + 43*sqrt(2)). - Simon Plouffe, Mar 04 2021

A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -2, 0, 1, -1, 0, 0, 2, 0, 1, -1, 0, -1, 0, -3, 0, 1, -1, 0, -1, 2, -1, 4, 0, 1, -1, 0, -1, 1, -2, 1, -5, 0, 1, -1, 0, -1, 1, 0, 1, -1, 6, 0, 1, -1, 0, -1, 1, -1, 0, -2, 1, -8, 0, 1, -1, 0, -1, 1, -1, 2, -1, 4, 0, 10, 0, 1, -1, 0, -1, 1, -1, 1, -2, 1, -4, 0, -12, 0

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

			Square array begins:
1,  1,  1,  1,  1,  1,  ...
0, -1, -1, -1, -1, -1,  ...
0,  1,  0,  0,  0,  0,  ...
0, -2,  0, -1, -1, -1,  ...
0,  2,  0,  2,  1,  1,  ...
0, -3, -1, -2,  0, -1,  ...

Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.
Main diagonal gives A081362.
Cf. A286653, A286656, A290307.

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

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

For asymptotics of column k see comment from Vaclav Kotesovec in A145707.

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A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.

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A134747 Expansion of q * (chi(-q) / chi(-q^4))^8 in powers of q where chi() is a Ramanujan theta function.

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A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

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