A008438 -id:A008438 - cp's OEIS frontend

A341021 Number of partitions of n into 4 distinct nonzero triangular numbers.

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 1, 1, 0, 2, 1, 3, 0, 1, 2, 0, 2, 2, 1, 3, 2, 0, 2, 2, 2, 1, 3, 0, 4, 3, 1, 3, 2, 2, 3, 2, 1, 5, 3, 3, 2, 4, 1, 2, 5, 1, 5, 3, 2, 5, 3, 3, 4, 4, 3, 4, 6, 0, 6, 4, 2, 7, 4, 3, 5, 4, 3, 5, 5, 5, 4, 5, 5, 5, 8, 2, 6, 5, 1, 10, 5, 4, 7, 7, 4

Offset: 20

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A008438, A010054, A307597, A307598, A319814, A340949, A341022, A341023, A341024, A341025, A341026, A341027.

A247821 Numbers k such that sigma(sigma(2k-1)) is a prime p.

Original entry on oeis.org

2, 1334, 1969, 28669, 86006, 126961, 338603654, 536801281, 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985

Offset: 1

Jaroslav Krizek, Sep 28 2014

Numbers n such that A000203(A000203(2n-1)) = A000203(A008438(n-1)) = A051027(2n-1) is a prime p.
Corresponding values of primes p are 7, 8191, 8191, 131071, 524287, 524287, ... (= A247822). Conjecture: The primes p are Mersenne primes (A000668).
sigma(sigma(2*a(9)-1)) > 10^16.
If the above conjecture is true, the next terms are 366479720500691270, 375344017599431990, 500461553802019261, 554079264075351985, 98375588019240949991670086, ... . - Hiroaki Yamanouchi, Oct 01 2014
a(13) > 5*10^18. - Giovanni Resta, Feb 14 2020

			Number 1334 is in sequence because sigma(sigma(2*1334-1)) = sigma(sigma(2667)) = sigma(4096) = 8191, i.e., prime.

Cf. A000203, A008438, A247790, A247791, A247820, A247822, A247823, A247954.

[n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1)))]

Select[Range[10^6], PrimeQ[DivisorSigma[1, DivisorSigma[1, 2 # - 1]]] &] (* Robert Price, May 17 2019 *)

for(n=1,10^7,if(ispseudoprime(sigma(sigma(2*n-1))),print1(n,", "))) \\ Derek Orr, Sep 29 2014

a(n) = (A247838(n) +1) / 2.

a(n)-1 = numbers n such that sigma(sigma(2n+1)) is a prime p: 1, 1333, 1968, 28668, 86005, 126960, ...

a(7)-a(8) from Hiroaki Yamanouchi, Oct 01 2014

a(9)-a(12) from Giovanni Resta, Feb 14 2020

A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 6, -8, 13, -12, 14, -24, 18, -20, 32, -24, 31, -40, 30, -32, 48, -48, 38, -56, 42, -44, 78, -48, 57, -72, 54, -72, 80, -60, 62, -104, 84, -68, 96, -72, 74, -124, 96, -80, 121, -84, 108, -120, 90, -112, 128, -120, 98, -156, 102, -104, 192, -108, 110

Offset: 0

Michael Somos, Aug 10 2006

sign

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

Number 33 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 13*x^4 - 12*x^5 + 14*x^6 - 24*x^7 + ...
G.f. = q - 4*q^3 + 6*q^5 - 8*q^7 + 13*q^9 - 12*q^11 + 14*q^13 - 24*q^15 + ...

J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.179
Hardy, et al., Collected Papers of Srinivasa Ramanujan, p. 326, Question 359.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001938, A008438, A097723, A112610, A134343, A258831, A258835.

A := Basis( ModularForms( Gamma0(16), 2), 110); A[2] - 4*A[4]; /* Michael Somos, Jun 10 2015 */

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^2 / (4 q^(1/2)), {q, 0, n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^4, {q, 0, n}]; (* Michael Somos, Oct 14 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[1, 2 n + 1]]; (* Michael Somos, Jun 15 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sigma(2*n + 1))};

A = ModularForms( Gamma0(16), 2, prec=110).basis(); A[1] - 4*A[3]; # Michael Somos, Jun 27 2013

Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2))^4 in powers of q.
Expansion of q^(-1/2)/4 * k * k' * (K / (Pi/2))^2 in powers of q where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^n, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (-1)^e * (p^(e+1) - 1) / (p - 1) if p == 3 (mod 4).
Given g.f. A(x), then B(x) = 4 * Integral_{0..x} A(x^2) dx = arcsin(4 * x * A001938(x^2)) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = sin(u + v) / 2 - sin((u - v) / 2). - Michael Somos, Oct 14 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 27 2013
G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)))^4.
G.f.: Sum_{k>0} -(-1)^k * (2*k - 1) * x^(k - 1) / (1 + x^(2*k - 1)).
G.f.: (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(4*k)))^4.
G.f.: (Sum_{k>0} (-1)^floor(k/2) * x^((k^2 - k)/2))^4.
G.f.: Sum_{k>0} (-1)^k * (2*k - 1) * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A008438(n). a(2*n) = A112610(n). a(2*n + 1) = -4 * A097723(n).
Convolution square of A134343. - Michael Somos, Jun 20 2012
a(3*n + 2) = 6 * A258831(n). a(4*n + 3) = -8 * A258835(n). - Michael Somos, Jun 11 2015

A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

Original entry on oeis.org

1, 5, 4, 13, 6, 20, 8, 29, 13, 30, 12, 52, 14, 40, 24, 61, 18, 65, 20, 78, 32, 60, 24, 116, 31, 70, 40, 104, 30, 120, 32, 125, 48, 90, 48, 169, 38, 100, 56, 174, 42, 160, 44, 156, 78, 120, 48, 244, 57, 155, 72, 182, 54, 200, 72, 232, 80, 150, 60, 312, 62, 160

Offset: 1

Michel Lagneau, Aug 28 2012

Multiplicative because a(n) = -A002129(2*n), A002129 is multiplicative and a(1) = -A002129(2) = 1. - Andrew Howroyd, Jul 31 2018

			a(6) = 20 because the divisors of 2*6 = 12 are {1, 2, 3, 4, 6, 12} and (12 + 6 + 4 +2) - (3 + 1) = 20.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Hartosh Singh Bal and Gaurav Bhatnagar, Two curious congruences for the sum of divisors function, arXiv:2102.10804 [math.NT], 2021. Mentions this sequence.
H. Movasati and Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2017.

Cf. A000593, A002129, A022998 (Moebius transform), A074400, A195382, A195690.

Cf. A002513, A062731, A111003, A239050.

with(numtheory):for n from 1 to 100 do:x:=divisors(2*n):n1:=nops(x):s0:=0:s1:=0:for m from 1 to n1 do: if irem(x[m],2)=0 then s0:=s0+x[m]:else s1:=s1+x[m]:fi:od:if s0>s1  then printf(`%d, `,s0-s1):else fi:od:

a[n_] := DivisorSum[2n, (1 - 2 Mod[#, 2]) #&];
Array[a, 62] (* Jean-François Alcover, Sep 13 2018 *)
edod[n_]:=Module[{d=Divisors[2n]},Total[Select[d,EvenQ]]-Total[ Select[ d,OddQ]]]; Array[edod,70] (* Harvey P. Dale, Jul 30 2021 *)

a(n) = 4*sigma(n) - sigma(2*n); \\ Andrew Howroyd, Jul 28 2018

From Andrew Howroyd, Jul 28 2018: (Start)
a(n) = 4*sigma(n) - sigma(2*n).
a(n) = -A002129(2*n). (End)
G.f.: Sum_{k>=1} x^k*(1 + 4*x^k + x^(2*k))/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Sep 14 2019
a(p) = p + 1 for p prime >= 3. - Bernard Schott, Sep 14 2019
a(n) = A239050(n) - A062731(n) - Omar E. Pol, Mar 06 2021 (after Andrew Howroyd)
From Amiram Eldar, Nov 18 2022: (Start)
Multiplicative with a(2^e) = 2^(e+2) - 3, and a(p^e) = sigma(p^e) = (p^(e+1) - 1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1+2^(1-s)). - Amiram Eldar, Jan 05 2023
From Peter Bala, Sep 25 2023: (Start)
a(2*n) = sigma(2*n) + 2*sigma(n); a(2*n+1) = sigma(2*n+1) = A008438(n)
G.f.: A(q) = Sum_{n >= 1} n*q^n*(1 + 3*q^n)/(1 - q^(2*n)).
Logarithmic g.f.: Sum_{n >= 1} a(n)*q^n/n = Sum_{n >= 1} log(1/(1 - q^n)) + Sum_{n >= 1} log(1/(1 - q^(2*n))) = log (G(q)), where G(q) is the g.f. of A002513. (End)

A247822 Corresponding values of primes p from A247821 and A247838.

Original entry on oeis.org

7, 8191, 8191, 131071, 524287, 524287, 2147483647, 2147483647, 2305843009213693951, 2305843009213693951, 2305843009213693951, 2305843009213693951

Offset: 1

Jaroslav Krizek, Sep 28 2014

Conjecture: all terms are Mersenne primes (A000668).

Conjecture: next terms are 2305843009213693951, 2305843009213693951, 2305843009213693951, 2305843009213693951 and 618970019642690137449562111. - Jaroslav Krizek, Mar 25 2015

			a(2) = 8191 because sigma(sigma(2*A247821(2)-1)) = sigma(sigma(A247838(2))) = 8191.

Cf. A000203, A008438, A247790, A247791, A247820, A247821, A247823, A247954.

[SumOfDivisors(SumOfDivisors(n)): n in [A247838(n)]];

a(n) = sigma(sigma(2*A247821(n)-1)) = A000203(A000203(2*A247821(n)-1)) = A051027(2*A247821(n)-1).

a(n) = sigma(sigma(A247838(n))) = A000203(A000203(A247838(n))) = A051027(A247838(n)).

a(7)-a(8) from Jaroslav Krizek, Mar 25 2015

a(9)-a(12) from Giovanni Resta, Feb 14 2020

A229615 Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 3, 0, -1, 0, 7, -8, 6, 0, 1, 0, 8, -12, 15, 0, -7, 0, 18, -16, 12, 0, 5, 0, 14, -26, 24, 0, -6, 0, 31, -24, 18, 0, -5, 0, 20, -28, 42, 0, -8, 0, 36, -48, 24, 0, 13, 0, 31, -36, 42, 0, -25, 0, 56, -40, 30, 0, 6, 0, 32, -64, 63, 0, -12, 0, 54, -48, 48, 0

Offset: 2

Michael Somos, Sep 26 2013

sign

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

			G.f. = q^2 - 2*q^3 + 3*q^4 - q^6 + 7*q^8 - 8*q^9 + 6*q^10 + q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 2..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008438, A033686, A093829, A144614, A229616.

Basis( ModularForms( Gamma0(6), 2), 70)[3] /* Michael Somos, Mar 05 2023 */

a[ n_] := If[n < 1, 0, Sum[ {0, 1, -2, 1, 0, 0}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, 0, Sum[ {0, 1/2, -2/3, 1/2, 0, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(3/2)]^6 / EllipticTheta[ 2, 0, q^(1/2)]^2 / 16, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv( n, d, n/d * [0, 0, 1, -2, 1, 0][d%6 + 1]))};

{a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 1/2, -2/3, 1/2, 0][d%6 + 1]))};

{a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3))^2, n))};

ModularForms( Gamma0(6), 2, prec=70).2;

Expansion of (a(q) - a(q^2))^2 / 36 = c(q^2)^4 / (9 * c(q)^2) in powers of q where a(), c() are cubic AGM theta functions.
Expansion of ((eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3))^2 in powers of q.
Euler transform of period 6 sequence [ -2, 2, 4, 2, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/12) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229616.
G.f.: sum_{k>0} x^(6*k-4) / (1 - x^(6*k-4))^2 - 2 * x^(6*k-3) / (1 - x^(6*k-3))^2 + x^(6*k-2) / (1 - x^(6*k-2))^2.
G.f.: sum_{k>0} (3*k-2) * x^(6*k-4) / (1 - x^(6*k-4)) - (4*k-2) * x^(6*k-3) / (1 - x^(6*k-3)) + (3*k-1) * x^(6*k-2) / (1 - x^(6*k-2)).
a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 2) = A144614(n). a(6*n + 3) = -2 * A008438(n). a(6*n + 4) = 3 * A033686(n).
Convolution square of A093829.

A247790 Primes p such that sigma(sigma(2p-1)) is a prime.

Original entry on oeis.org

2, 28669, 126961, 500461553802019261

Offset: 1

Jaroslav Krizek, Sep 28 2014

The next term, if it exists, must be greater than 5*10^7.
Primes p such that A247954(p) = A000203(A000203(2p-1)) = A000203(A008438(p-1)) = A051027(2p-1) is a prime q. The corresponding values of the primes q are: 7, 131071, 524287, ... (A247791). Conjecture: the primes q are Mersenne primes (A000668).
Conjecture: the next term is 500461553802019261 (see comment from Hiroaki Yamanouchi in A247821). - Jaroslav Krizek, Oct 08 2014
These are the primes in A247821. - M. F. Hasler, Oct 14 2014
No other terms up to 5*10^10. - Michel Marcus, Feb 11 2020
a(5) > 5*10^18. - Giovanni Resta, Feb 14 2020

			Prime 2 is in the sequence because sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7, i.e., prime.

Cf. A000203, A008438, A247791, A247820, A247821, A247822, A247823, A247954.

[p: p in PrimesUpTo(50000000) | IsPrime(SumOfDivisors(SumOfDivisors(2*p-1)))]

with(numtheory): A247790:=n->`if`(isprime(n) and isprime(sigma(sigma(2*n-1))),n,NULL): seq(A247790(n), n=1..130000); # Wesley Ivan Hurt, Oct 17 2014

forprime(p=1,10^7,if(ispseudoprime(sigma(sigma(2*p-1))),print1(p,", "))) \\ Derek Orr, Sep 29 2014

a(4) from Giovanni Resta, Feb 14 2020

A247791 Primes p such that there is a prime q for which sigma(sigma(2*q-1)) = p.

Original entry on oeis.org

7, 131071, 524287

Offset: 1

Jaroslav Krizek, Sep 28 2014

The next term, if it exists, must be greater than 5*10^7.
Primes p such that there is prime q for which sigma(sigma(2*q-1)) = A247954(q) = A000203(A000203(2*q-1)) = A000203(A008438(q-1)) = A051027(2*q-1) = p.
Corresponding values of primes q: 2, 28669, 126961, ... (A247790).
Conjecture: Subsequence of Mersenne primes.
Conjecture: the next term is 2305843009213693951 when 2305843009213693951 = sigma(sigma(2*500461553802019261-1)) where 500461553802019261 is prime (see comment of Hiroaki Yamanouchi in A247821). - Jaroslav Krizek, Oct 08 2014

			Prime 7 is in sequence because there is prime 2 such that sigma(sigma(2*2-1)) = sigma(sigma(3)) = sigma(4) = 7.

Cf. A000203, A008438, A247790, A247820, A247821, A247822, A247823, A247954.

[SumOfDivisors(SumOfDivisors(2*n-1)): n in [A247790(n)]];

[SumOfDivisors(SumOfDivisors(2*n-1)): n in[1..1000000] | IsPrime(SumOfDivisors(SumOfDivisors(2*n-1))) and IsPrime(n)];

forprime(p=1,10^7,if(ispseudoprime(sigma(sigma(2*p-1))),print1(sigma(sigma(2*p-1)),", "))) \\ Derek Orr, Sep 29 2014

A247823 Mersenne primes p such that there is a number k with sigma(sigma(2k-1)) = p.

Original entry on oeis.org

7, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111

Offset: 1

Jaroslav Krizek, Sep 28 2014

Mersenne primes p such that there is a number m such that sigma(sigma(m)) = p.
Distinct values attained by the A247822(n) function, in ascending order.
Mersenne primes p such that there are a numbers n and m such that sigma(sigma(2n-1)) = sigma(sigma(2*A247821(n)-1)) = A000203(A000203(2*A247821(n)-1)) = A051027(2*A247821(n)-1) = sigma(sigma(A247838(m))) = A000203(A000203(A247838(m))) = A051027(A247838(m)) where m = 2n-1.
The Mersenne prime 7 is the only prime p such that there is a prime q with sigma(sigma(q)) = p.

			Mersenne prime 8191 is in sequence because there are numbers n = 1334 and 1969 with sigma(sigma(2*n-1)) = 8191.

Cf. A000203, A008438, A247790, A247791, A247820, A247821, A247822, A247954.

Cf. A000668 (Mersenne primes).

Set(Sort([SumOfDivisors(SumOfDivisors(n)): n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(n)))])) // Jaroslav Krizek, Mar 25 2015

a(5)-a(7) from Jaroslav Krizek, Mar 25 2015

A005879 Theta series of D_4 lattice with respect to deep hole.

Original entry on oeis.org

8, 32, 48, 64, 104, 96, 112, 192, 144, 160, 256, 192, 248, 320, 240, 256, 384, 384, 304, 448, 336, 352, 624, 384, 456, 576, 432, 576, 640, 480, 496, 832, 672, 544, 768, 576, 592, 992, 768, 640, 968, 672, 864, 960, 720, 896, 1024, 960, 784, 1248, 816, 832, 1536

Offset: 0

N. J. A. Sloane

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

The D_4 lattice is the set of all integer quadruples [a, b, c, d] where a + b + c + d is even. The deep holes are quadruples [a, b, c, d] where each coordinate is half an odd integer and where a + b + c + d is even. - Michael Somos, May 23 2012

			8 + 32*x + 48*x^2 + 64*x^3 + 104*x^4 + 96*x^5 + 112*x^6 + 192*x^7 + ...
8*q + 32*q^3 + 48*q^5 + 64*q^7 + 104*q^9 + 96*q^11 + 112*q^13 + ...
.
For n = 2 the objects counted are the ways to represent the integer 5 = (2*n+1) as a sum of 4 squares, 0 and negative numbers allowed.
[-2,-1,0,0], [-2,0,-1,0], [-2,0,0,-1], [-2,0,0,1], [-2,0,1,0], [-2,1,0,0],
[-1,-2,0,0], [-1,0,-2,0], [-1,0,0,-2], [-1,0,0,2], [-1,0,2,0], [-1,2,0,0],
[0,-2,-1,0], [0,-2,0,-1], [0,-2,0,1], [0,-2,1,0], [0,-1,-2,0], [0,-1,0,-2],
[0,-1,0,2], [0,-1,2,0], [0,0,-2,-1], [0,0,-2,1], [0,0,-1,-2], [0,0,-1,2],
[0,0,1,-2], [0,0,1,2], [0,0,2,-1], [0,0,2,1], [0,1,-2,0], [0,1,0,-2],
[0,1,0,2], [0,1,2,0], [0,2,-1,0], [0,2,0,-1], [0,2,0,1], [0,2,1,0],
[1,-2,0,0], [1,0,-2,0], [1,0,0,-2], [1,0,0,2], [1,0,2,0], [1,2,0,0],
[2,-1,0,0], [2,0,-1,0], [2,0,0,-1], [2,0,0,1], [2,0,1,0], [2,1,0,0].
- _Peter Luschny_, Nov 03 2015

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to D_4 lattice

Cf. A000118, A005880, A008438, A096727.

S:= series(JacobiTheta2(0,q)^4/(2*q), q, 202):
seq(coeff(S,q,2*j),j=0..100); # Robert Israel, Nov 03 2015

(* a(n) gives the number of ways to represent the integer 2n+1 as a sum of 4 squares *) a[n_] := SquaresR[4, 2n+1]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Nov 03 2015 *)
terms = 53; QP = QPochhammer; s = 8 QP[q^2]^8/QP[q]^4 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *)

{a(n) = if( n<0, 0, 8 * sigma(2*n + 1))} /* Michael Somos, Apr 11 2004 */

q='q+O('q^66); Vec(8*(eta(q^2)^2/eta(q))^4) \\ Joerg Arndt, Nov 03 2015

Expansion of Jacobi theta_2(q)^4/(2q) in powers of q^2. - Michael Somos, Apr 11 2004
Expansion of q^(-1/2) * 8 * (eta(q^2)^2 / eta(q))^4 in powers of q. - Michael Somos, Apr 11 2004
Expansion of 8 * psi(x)^4 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, May 23 2012
Expansion of (phi(q)^4 - phi(-q)^4) / (2 * q) in powers of q^2. - Michael Somos, May 23 2012
G.f.: 8 * (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004
a(n) = 8 * A008438(n) = 4 * A005880(n) = A000118(2*n + 1) = - A096727(2*n + 1). - Michael Somos, Nov 01 2006

cp's OEIS Frontend

A341021 Number of partitions of n into 4 distinct nonzero triangular numbers.

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A247821 Numbers k such that sigma(sigma(2k-1)) is a prime p.

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A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

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A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

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A247822 Corresponding values of primes p from A247821 and A247838.

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A229615 Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function.

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A247790 Primes p such that sigma(sigma(2p-1)) is a prime.

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