A115110 -id:A115110 - cp's OEIS frontend

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

1, 2, 1, 2, 2, 0, 3, 2, 0, 2, 2, 2, 1, 2, 0, 2, 4, 0, 2, 0, 1, 4, 2, 0, 2, 2, 0, 2, 2, 2, 1, 4, 0, 0, 2, 0, 4, 2, 2, 2, 0, 0, 3, 2, 0, 2, 4, 0, 2, 2, 0, 4, 0, 0, 0, 4, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 4, 2, 0, 2, 2, 0, 3, 2, 0, 0, 4, 0, 2, 2, 0, 6, 0, 2, 2, 0, 0, 2, 2, 0, 1, 4, 2, 2, 4, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present sequence gives the expansion coefficients of psi(q)^2.

Also the number of positive odd solutions to equation x^2 + y^2 = 8*n + 2. - Seiichi Manyama, May 28 2017

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 3*x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^11 + ...
G.f. for B(q) = q * A(q^4) = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag. See p. 139 Example (iv).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
R. W. Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, in Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. See p. 279.
R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. [See Pi_q.]
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916. See vol. 2, p 31, Article 272.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 165.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 72, Eq (31.2); p. 78, Eq. following (32.25).
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. See p. 108.
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp. 73-94.
Hjalmar Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113414, A113446, A113652 all satisfy A(4n+1)=a(n).
Cf. A004020, A005883, A104794, A052343, A199015 (partial sums).
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A002175, A053692, A113407, A121444.
Cf. A000217, A001938, A004018.
Cf. A274621 (reciprocal series).
Cf. A001826, A001842.
Cf. A019669, A079006.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A := Basis( ModularForms( Gamma1(8), 1), 420); A[2]; /* Michael Somos, Jan 31 2015 */

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A002654 := proc(n) sigmamr(n,4,1)-sigmamr(n,4,3) ; end proc:
A008441 := proc(n) A002654(4*n+1) ; end proc:
seq(A008441(n),n=0..90) ; # R. J. Mathar, Mar 23 2011

Plus@@((-1)^(1/2 (Divisors[4#+1]-1)))& /@ Range[0, 104] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q], {q, 0, n + 1/4}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q]^2, {q, 0, 2 n + 1/2}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]];  (* Michael Somos, Jun 08 2014 *)
QP = QPochhammer; s = QP[q^2]^4/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
TriangleQ[n_] := IntegerQ@Sqrt[8n +1]; Table[Count[FrobeniusSolve[{1, 1}, n], {?TriangleQ}], {n, 0, 104}] (* Robert G. Wilson v, Apr 15 2017 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k * (k+1)/2), x * O(x^n))^2, n) )};

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, (-1)^(d\2)))}; /* Michael Somos, Sep 02 2005 */

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

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 14 2005 */

{ my(q='q+O('q^166)); Vec(eta(q^2)^4 / eta(q)^2) } \\ Joerg Arndt, Apr 16 2017

ModularForms( Gamma1(8), 1, prec=420).1; # Michael Somos, Jun 08 2014

This sequence is the quadrisection of many sequences. Here are two examples:
a(n) = A002654(4n+1), the difference between the number of divisors of 4*n+1 of form 4*k+1 and the number of form 4*k-1. - David Broadhurst, Oct 20 2002
a(n) = b(4*n + 1), where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Sep 14 2005
G.f.: (Sum_{k>=0} x^((k^2 + k)/2))^2 = (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)).
Expansion of Jacobi theta (theta_2(0, sqrt(q)))^2 / (4 * q^(1/4)).
Sum[d|(4n+1), (-1)^((d-1)/2) ].
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 4 * v * w^2 - u^2 * w. - Michael Somos, Sep 14 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1 * u3 - (u2 - u6) * (u2 + 3*u6). - Michael Somos, Sep 14 2005
Expansion of Jacobi k/(4*q^(1/2)) * (2/Pi)* K(k) in powers of q^2. - Michael Somos, Sep 14 2005. Convolution of A001938 and A004018. This appears in the denominator of the Jacobi sn and cn formula given in the Abramowitz-Stegun reference, p. 575, 16.23.1 and 16.23.2, where m=k^2. - Wolfdieter Lang, Jul 05 2016
G.f.: Sum_{k>=0} a(k) * x^(2*k) = Sum_{k>=0} x^k / (1 + x^(2*k + 1)).
G.f.: Sum_{k in Z} x^k / (1 - x^(4*k + 1)). - Michael Somos, Nov 03 2005
Expansion of psi(x)^2 = phi(x) * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Jan 25 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A104794.
Euler transform of period 2 sequence [ 2, -2, ...].
G.f.: q^(-1/4) * eta(q^2)^4 / eta(q)^2. See also the Fine reference.
a(n) = Sum_{k=0..n} A010054(k)*A010054(n-k). - Reinhard Zumkeller, Nov 03 2009
A004020(n) = 2 * a(n). A005883(n) = 4 * a(n).
Convolution square of A010054.
G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^(2*k-1))^2.
a(2*n) = A113407(n). a(2*n + 1) = A053692(n). a(3*n) = A002175(n). a(3*n + 1) = 2 * A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jun 08 2014
G.f.: exp( Sum_{n>=1} 2*(x^n/n) / (1 + x^n) ). - Paul D. Hanna, Mar 01 2016
a(n) = A001826(2+8*n) - A001842(2+8*n), the difference between the number of divisors 1 (mod 4) and 3 (mod 4) of 2+8*n. See the Ono et al. link, Corollary 1, or directly the Niven et al. reference, p. 165, Corollary (3.23). - Wolfdieter Lang, Jan 11 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 - x^1)^2 / (1 - x^3 + x^2*(1 - x^2)^2 / (1 - x^5 + x^3*(1 - x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
Given g.f. A(x), and B(x) is the g.f. for A079006, then B(x) = A(x^2) / A(x) and B(x) * B(x^2) * B(x^4) * ... = 1 / A(x). - Michael Somos, Apr 20 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) - x^(2*k))^(n-k) = Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) + x^(2*k))^(n-k) * (-1)^k = Sum_{n>=0} a(n)*x^(2*n). (End)
From Peter Bala, Jan 05 2021: (Start)
G.f.: Sum_{n = -oo..oo} x^(4*n^2+2*n) * (1 + x^(4*n+1))/(1 - x^(4*n+1)). See Agarwal, p. 285, equation 6.20 with i = j = 1 and mu = 4.
For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/4) = a(n).
If n > 0 and p are coprime then a(n*p + (p^2 - 1)/4) = 0. The proofs are similar to those given for the corresponding results for A115110. Cf. A000729.
For prime p of the form 4*k + 1 and for n not congruent to (p - 1)/4 (mod p) we have a(n*p^2 + (p^2 - 1)/4) = 3*a(n) (since b(n), where b(4*n+1) = a(n), is multiplicative). (End)
From Peter Bala, Mar 22 2021: (Start)
G.f. A(q) satisfies:
A(q^2) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+2)) (set z = q, alpha = q^2, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)) (set z = q^2, alpha = q, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^n/(1 + q^(2*n+1))^2 = Sum_{n = -oo..oo} q^(3*n+1)/(1 + q^(2*n+1))^2. (End)
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) + 2 * Sum_{n>=1, k>=0} (-1)^k * q^(k*(k+2*n+1)+n). - Mamuka Jibladze, May 17 2021
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) * (1 + q^(2*k+1))/(1 - q^(2*k+1)). - Mamuka Jibladze, Jun 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Oct 15 2022

More terms and information from Michael Somos, Mar 23 2003

A282937 a(n) = A000728(5*n).

Original entry on oeis.org

1, -6, 9, 10, -30, 1, 5, 51, 10, -100, 20, -55, 109, 110, -130, -1, -110, 160, 10, -230, 100, 15, 191, 120, -230, -100, -89, 160, 90, -340, 120, 5, 300, 200, -260, -1, -275, 240, -100, -270, 119, -165, 260, 410, -200, -40, 20, 200, -110, -500, 180, -54, 140

Offset: 0

Seiichi Manyama, Feb 25 2017

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This generalized function is related to two following identities; R(q^5) - q - q^2/R(q^5) = (q; q){infinity}/(q^25; q^25){infinity}, R^5(q^5) - 11*q^5 - q^10/R^5(q^5) = ((q^5; q^5){infinity}/(q^25; q^25){infinity})^6, where R(q) is the Rogers-Ramanujan continued function and (q; q)_n is the q-Pochhammer symbol. See the reference.

			G.f.: 1 - 6*q + 9*q^2 + 10*q^3 - 30*q^4 + q^5 + 5*q^6 + 51*q^7 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 185.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A000728, A182821 (Product_{n>=1} (1 - q^(5*n))/(1 - q^n)^6), A282941.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), this sequence (k=5), A282942 (k=7).

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018

CoefficientList[Series[Product[(1 - x^j)^6/(1 - x^(5*j)), {j,1,62}], {x,0,60}], x] (* G. C. Greubel, Nov 18 2018 *)

m=60; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^j)^6/(1 - x^(5*j)))) \\ G. C. Greubel, Nov 18 2018

R = PowerSeriesRing(ZZ, 'x')
prec = 60
x = R.gen().O(prec)
s = prod((1 - x^j)^6/(1 - x^(5*j)) for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

G.f.: Product_{n>=1} (1 - q^n)^6/(1 - q^(5*n)).
a(n) = (-1)^j mod 5 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 5.
Sum_{k=0..n} a(k)*A182821(n-k) = 0 for n > 0. - Seiichi Manyama, Feb 28 2017
G.f.: exp( Sum_{n>=1} -sigma(5*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(5*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 1, -133, -140, 308, -70, 174, 56, -518, -141, -63, 868, -140, 238, 294, -1029, -1154, -203, 2366, -658, 1296, 350, -1547, -1295, -1666, 3234, -2128, 2534, 2464, -2577, -3087, -609, 5600, -2716, 2435, 294, -3745, -4249, -1015, 8526

Offset: 0

Seiichi Manyama, Feb 25 2017

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			G.f.: 1 - 8*q + 20*q^2 - 70*q^4 + 64*q^5 + 56*q^6 + q^7 - 133*q^8 - 140*q^9 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 192.

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

Cf. A282941.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), A282937 (k=5), this sequence (k=7).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^j)^8/(1-x^(7*j)): j in [1..m]]) )); // G. C. Greubel, Nov 18 2018

nmax = 50; CoefficientList[Series[Product[(1-x^k)^8/(1-x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Nov 18 2018 *)

m=50; x='x+O('x^m); Vec(prod(j=1,m, (1-x^j)^8/(1-x^(7*j)))) \\ G. C. Greubel, Nov 18 2018

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = prod((1-x^j)^8/(1-x^(7*j)) for j in (1..50))
list(s) # G. C. Greubel, Nov 18 2018

G.f.: exp( Sum_{n>=1} -sigma(7*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017

a(n) = -(1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

Original entry on oeis.org

1, -4, 2, 9, -9, -2, 0, -5, 9, 9, 0, -9, -1, -9, 0, -1, 9, 9, -9, 9, 0, 9, -5, -18, -18, 9, 7, 0, 9, 0, 0, 9, 9, -18, 18, -7, -9, -9, -9, 9, -4, -9, -9, 18, 9, 0, 18, 9, 0, -9, -9, -8, -9, 18, -9, 9, -18, 1, -9, -18, 9, 0, 18, 18, 0, 0, 9, -9, 18, -9, 5, -9, 0, -9, -9, -9, -18, 11, 9

Offset: 0

Paul D. Hanna, Feb 16 2011

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			G.f. = 1 - 4*x + 2*x^2 + 9*x^3 - 9*x^4 - 2*x^5 - 5*x^7 + 9*x^8 + ... - _Michael Somos_, Jul 12 2018

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

Cf. A182819, A144613, A000203.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), this sequence (k=3), A282937 (k=5), A282942 (k=7).

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 12 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n,-sigma(3*m)*x^m/m)+x*O(x^n)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^4/eta(X^3),n)}

G.f.: E(x)^4/E(x^3) where E(x) = Product_{n>=1} (1-x^n). [From a formula by Joerg Arndt in A182819]
a(n) = -(1/n)*Sum_{k=1..n} sigma(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017
Expansion of E(x) * E(x*w) * E(x/w) in powers of x^3 where w = exp(2 Pi i / 3). - Michael Somos, Jul 12 2018

A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -12, 58, -133, 95, 194, -418, 97, 325, -99, -238, 169, -217, 131, 190, -145, 441, -647, 169, -527, 72, 1129, 313, -972, 2, -491, -565, 1944, -1175, -216, 972, 863, -1259, 288, 0, -1155, -1355, -207, 2925, 1753, 1402, -2387, -2257, -1030, 315, 432, -72, 1621, 358

Offset: 0

Seiichi Manyama, Mar 02 2017

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

Cf. A224613 (sigma(6*n)), A283119 (exp( Sum_{n>=1} sigma(6*n)*x^n/n )).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), this sequence (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

G.f.: Product_{n>=1} (1 - x^n)^12 * (1 - x^(6*n))/((1 - x^(2*n))^4 * (1 - x^(3*n))^3).

a(n) = -(1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283168 Expansion of exp( Sum_{n>=1} -sigma(8n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -15, 97, -350, 770, -1133, 1540, -2731, 4230, -3960, 3402, -6580, 9167, -5390, 4310, -11061, 12320, -5306, 2030, -7530, 14784, -4340, -10119, -9240, 20090, 11438, -17275, -4928, 2270, 14080, -26840, 7700, 16646, 24640, -53760, 7449, 10780, 46200, -61600

Offset: 0

Seiichi Manyama, Mar 02 2017

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

Cf. A283120 (exp( Sum_{n>=1} sigma(8*n)*x^n/n )), A283122 (sigma(8*n)).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), this sequence (k=8), A283169 (k=9).

G.f.: Product_{n>=1} (1 - x^n)^15/(1 - x^(2*n))^7.

a(n) = -(1/n)*Sum_{k=1..n} sigma(8*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283169 Expansion of exp( Sum_{n>=1} -sigma(9n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -13, 65, -126, -117, 988, -1377, -1157, 5382, -4419, -4212, 12519, -11179, -5058, 27378, -23005, -16488, 44343, -30249, -18513, 73710, -56259, -38741, 93483, -69570, -23778, 137266, -90396, -74079, 140292, -108621, -39249, 222624, -145710, -99234

Offset: 0

Seiichi Manyama, Mar 02 2017

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

Cf. A283121 (exp( Sum_{n>=1} sigma(9*n)*x^n/n )), A283123 (sigma(9*n)).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), A283168 (k=8), this sequence (k=9).

G.f.: Product_{n>=1} (1 - x^n)^13/(1 - x^(3*n))^4.

a(n) = -(1/n)*Sum_{k=1..n} sigma(9*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283163 Expansion of exp( Sum_{n>=1} -sigma(4n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -7, 17, -14, 2, -21, 36, 13, -26, -24, 10, 12, -17, 34, 22, 19, -96, -10, 14, 38, 0, 12, -23, 72, -38, -2, -11, -64, -34, 0, 72, 84, -26, 0, 0, -79, 60, 24, -32, -58, -7, -84, 50, 26, 120, 0, 0, 46, -34, -64, 10, -119, 70, 0, 22, -70, 36, 37, -120, 0

Offset: 0

Seiichi Manyama, Mar 02 2017

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

Cf. A182820 (exp( Sum_{n>=1} sigma(4*n)*x^n/n )), A193553 (sigma(4*n)).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), this sequence (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

G.f.: Product_{n>=1} (1 - x^n)^7/(1 - x^(2*n))^3.

a(n) = -(1/n)*Sum_{k=1..n} sigma(4*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A185653 Expansion of exp( Sum_{n>=1} -3sigma(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -9, 30, -39, 0, 18, 49, 0, -192, 110, 81, 78, -130, 0, -30, -121, 0, 210, 320, -270, 0, -407, 0, 192, 190, 0, 0, 0, 351, -210, -418, 0, -510, 448, 0, 462, 611, 0, -960, 50, 0, 0, -350, 0, 450, -361, -162, 960, 0, 0, 798, -782, 0, -1170, -290, -441, 702, 850, 0, 0, 576

Offset: 0

Paul D. Hanna, Feb 16 2011

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When is a(n) zero (A258867)?

			G.f. = 1 - 9*x + 30*x^2 - 39*x^3 + 18*x^5 + 49*x^6 - 192*x^8 + 110*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Cf. A115110, A000203, A258867.

{a(n)=polcoeff(exp(sum(m=1,n,-3*sigma(2*m)*x^m/m)+x*O(x^n)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^9/eta(X^2)^3,n)}

Expansion of q^(-1/8)*eta(q)^9/eta(q^2)^3 in powers of q; equals the self-convolution cube of A115110 [See formula of Michael Somos for A115110].

A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -5, 2, 15, 12, -36, -92, -17, 167, 358, 283, -293, -1321, -2012, -1101, 2299, 7296, 10505, 6901, -7705, -31240, -52490, -51336, -6032, 91521, 217064, 303776, 250595, -36282, -575622, -1234465, -1684515, -1448538, -66980, 2610835, 6087681, 8990575

Offset: 0

Seiichi Manyama, Mar 03 2017

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

Cf. A283224 (exp( Sum_{n>=1} sigma_2(2*n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(2*n)*x^n/n ): A115110 (k=1), this sequence (k=2).
Cf. exp( Sum_{n>=1} -sigma_2(m*n)*x^n/n ): A073592 (m=1), this sequence (m=2), A283243 (m=3).

a(n) = -(1/n)*Sum_{k=1..n} sigma_2(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

cp's OEIS Frontend

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

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A282937 a(n) = A000728(5*n).

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A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.

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A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

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A283164 Expansion of exp( Sum_{n>=1} -sigma(6*n)*x^n/n ) in powers of x.

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A283168 Expansion of exp( Sum_{n>=1} -sigma(8*n)*x^n/n ) in powers of x.

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A283169 Expansion of exp( Sum_{n>=1} -sigma(9*n)*x^n/n ) in powers of x.

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A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.

A283168 Expansion of exp( Sum_{n>=1} -sigma(8n)x^n/n ) in powers of x.

A283169 Expansion of exp( Sum_{n>=1} -sigma(9n)x^n/n ) in powers of x.

A283163 Expansion of exp( Sum_{n>=1} -sigma(4n)x^n/n ) in powers of x.

A185653 Expansion of exp( Sum_{n>=1} -3sigma(2n)x^n/n ) in powers of x.

A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.