James Sellers - cp's OEIS frontend

A385520 Expansion of Product_{k>0} ((1 - q^(2k))(1 - q^(6k))^3)/((1 - q^k)(1 - q^(3k))(1 - q^(4k))(1 - q^(12*k))).

1, 1, 1, 3, 4, 5, 6, 9, 13, 16, 20, 27, 36, 44, 54, 69, 88, 107, 130, 162, 200, 240, 288, 351, 426, 507, 602, 723, 864, 1019, 1200, 1422, 1681, 1968, 2300, 2700, 3160, 3674, 4266, 4965, 5768, 6665, 7692, 8892, 10260, 11792, 13536, 15552, 17844, 20407

Offset: 0

James Sellers, Jul 01 2025

nonn

a(n) is the number of partitions of the integer n wherein each even part can appear at most twice, while each odd part can appear once, thrice, or four times.

Also, a(n) is the unsigned version of the sequence given in A293306 (this can be seen by replacing q by -q in the generating function).

			For n = 4, the a(4) = 4 partitions are 4, 3+1, 2+2, and 1+1+1+1.  Note that there is one other partition of 4 which is NOT counted by a(4); that is the partition 2+1+1. This partition is NOT counted by a(4) because the odd part 1 appears twice, and this is not allowed from the description given above.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 15.

Cf. A293306.

p:=product((1-q^(2*k))*(1-q^(6*k))^3/((1-q^k)*(1-q^(3*k))*(1-q^(4*k))*(1-q^(12*k))), k=1..1000): s:=series(p,q,1000):

nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k)) * (1 - x^(6*k))^3 / ((1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2025 *)

my(N=50,q='q+O('q^N)); Vec((eta(-q)*eta(-q^3))/eta(q^2)^2) \\ Joerg Arndt, Jul 02 2025

G.f.: Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).
G.f.: (eta(-q)*eta(-q^3))/eta(q^2)^2.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Jul 02 2025

A143981 The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).

Original entry on oeis.org

1, 3, 6, 9, 15, 19, 26, 36, 46, 59, 80, 100, 128, 167, 211, 267, 341, 429, 541, 682, 850, 1063, 1327, 1647, 2035, 2520, 3100, 3810, 4669, 5708, 6955, 8468, 10267, 12441, 15026, 18120, 21788, 26175, 31355, 37510, 44769, 53362, 63460, 75384, 89348

Offset: 1

Michael David Hirschhorn and James Sellers, Sep 06 2008

nonn

			For m = 4, the number of unigraphical partitions is A000041(4) + A001399(1) + A000005(5) + A083039(2) + 4 - 5 = 5 + 1 + 2 + 2 + 4 - 5 = 9.

S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Indust. Appl. Math., vol. 10 (1962), 496-506.
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. II. Uniqueness, J. Soc. Indust. Appl. Math., vol. 11 (1963), 135-147.

Cf. A000041, A001399, A000005, A083039.

with(combinat): with(numtheory): a:=proc(m) it:=round(m^2/12)+numbpart(m)+tau(m+1)+m-5: if m mod 6 = 0 then it:=it+2 fi: if m mod 6 = 1 then it:=it+1 fi: if m mod 6 = 2 then it:=it+3 fi: if m mod 6 = 3 then it:=it+1 fi: if m mod 6 = 4 then it:=it+2 fi: if m mod 6 = 5 then it:=it+2 fi: RETURN(it): end:

For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5.

A147877 The number of degree sequences with degree sum 2n representable by a non-separable graph (with multiple edges allowed).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 25, 43, 70, 113, 181, 283, 436, 666, 999, 1483, 2179, 3166, 4556, 6504, 9200, 12918, 18011, 24938, 34308, 46928, 63815, 86324, 116187, 155626, 207502, 275491, 364226, 479660, 629305, 822655, 1071694, 1391531, 1801041, 2323958, 2989883

Offset: 1

James Sellers, Nov 16 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
O. J. Rodseth, J. A. Sellers and H. Tverberg, Enumeration of the Degree Sequences of Non-Separable Graphs and Connected Graphs, European Journal of Combinatorics 30 (2009), 1301-1317.

Cf. A147878.

with(combinat): seq(numbpart(2*m) - numbpart(2*m - 1) - add(numbpart(j), j = 0 .. m-2), m=1..60);

a(n) = numbpart(2*n) - numbpart(2*n-1) - sum(j=0, n-2, numbpart(j)); \\ Michel Marcus, Nov 04 2016

a(n) = p(2n) - p(2n-1) - Sum_{j=0..n-2} p(j).
a(n) = A000041(2*n) - A000041(2*n-1) - A000070(n) + A000041(n) + A000041(n-1). - Vaclav Kotesovec, Nov 05 2016
a(n) ~ exp(2*Pi*sqrt(n/3))*Pi/(48*n^(3/2)) * (1 - (3*sqrt(3)/(2*Pi) + 13*Pi/(48*sqrt(3)))/sqrt(n)). - Vaclav Kotesovec, Nov 05 2016

Offset corrected by Michel Marcus, Nov 04 2016

A147878 The number of degree sequences with degree sum 2n representable by a connected graph (with multiple edges allowed).

Original entry on oeis.org

1, 2, 5, 11, 23, 46, 86, 156, 273, 463, 766, 1241, 1969, 3073, 4723, 7157, 10711, 15850, 23206, 33654, 48373, 68955, 97544, 137002, 191125, 264955, 365127, 500349, 682018, 924982, 1248502, 1677530, 2244229, 2989952, 3967732, 5245354, 6909211

Offset: 1

James Sellers, Nov 16 2008

nonn

			From _Gus Wiseman_, Oct 26 2018: (Start)
The a(1) = 1 through a(5) = 23 connected multigraphical partitions:
  (11)  (22)   (33)    (44)     (55)
        (211)  (222)   (332)    (433)
               (321)   (422)    (442)
               (2211)  (431)    (532)
               (3111)  (2222)   (541)
                       (3221)   (3322)
                       (3311)   (3331)
                       (4211)   (4222)
                       (22211)  (4321)
                       (32111)  (4411)
                       (41111)  (5221)
                                (5311)
                                (22222)
                                (32221)
                                (33211)
                                (42211)
                                (43111)
                                (52111)
                                (222211)
                                (322111)
                                (331111)
                                (421111)
                                (511111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
O. J. Rodseth, J. A. Sellers and H. Tverberg, Enumeration of the Degree Sequences of Non-Separable Graphs and Connected Graphs, European Journal of Combinatorics 30 (2009), 1301-1317.
Gus Wiseman, Connected multigraphs realizing each of the a(5) = 23 connected multigraphical graphical partitions of 10.

Cf. A000070, A000569, A007717, A096373, A147877, A209816, A320911, A320921 (no multiedges), A320923.

with(combinat): seq(numbpart(2*m) - numbpart(m - 1) - 2*add(numbpart(j), j = 0 .. m-2), m=1..60);

a(n) = numbpart(2*n) - numbpart(n-1) - 2*sum(j=0, n-2, numbpart(j)); \\ Michel Marcus, Nov 04 2016

a(n) = p(2n) - p(n-1) - 2*Sum_{j=0..n-2} p(j).
a(n) = A000041(2*n) - 2*A000070(n) + 2*A000041(n) + A000041(n-1). - Vaclav Kotesovec, Nov 05 2016
a(n) ~ exp(2*Pi*sqrt(n/3))/(8*sqrt(3)*n) * (1 - (sqrt(3)/(2*Pi) + Pi/(48*sqrt(3))) /sqrt(n)). - Vaclav Kotesovec, Nov 05 2016

Offset corrected by Michel Marcus, Nov 04 2016

A117574 Total number of permutations p of [n] such that |p(i+3) - p(i)| is not equal to 3 for 1 <= i <= n-3.

Original entry on oeis.org

1, 1, 2, 6, 20, 80, 384, 2240, 15424, 123456, 1110928, 11287232, 127016304, 1565107248, 20935873872, 301974271248, 4669727780624, 77046043259824, 1350585114106416, 25062108668100208, 490725684463001488, 10109820295907492304

Offset: 0

James Sellers, Apr 27 2006

a(n) is also number of ways to place n nonattacking pieces rook + leaper[3,3] on an n X n chessboard.

Vaclav Kotesovec, Table of n, a(n) for n = 0..30
Vaclav Kotesovec, Non-attacking chess pieces, Sixth edition, p. 633, Feb 02 2013.
Vaclav Kotesovec, Mathematica program for this sequence
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.

Cf. A002464, A110128.

Column k=3 of A333706.

Formula given in Tauraso reference.

Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 8/n + 30/n^2)/e^2.

Terms a(17)-a(28) from Vaclav Kotesovec, Apr 19 2011
Terms a(29)-a(30) from Vaclav Kotesovec, Apr 20 2012
a(0)=1 prepended by Alois P. Heinz, Feb 05 2023

A112603 Number of representations of n as the sum of a square and a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

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

			a(4) = 4 since we can write 4 = 2^2 + 0 = (-2)^2 + 0 = 1^2 + 3 = (-1)^2 + 3.
1 + 3*x + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + x^6 + 4*x^7 + 2*x^9 + 5*x^10 + ...
q + 3*q^9 + 2*q^17 + q^25 + 4*q^33 + 2*q^41 + q^49 + 4*q^57 + 2*q^73 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A139093.

a[n_] := DivisorSum[8n + 1, KroneckerSymbol[-2, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

{a(n) = if( n<0, 0, n = 8*n + 1; sumdiv( n, d, kronecker( -2, d)))} /* Michael Somos, Sep 29 2006 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 /(eta(x + A)^3 * eta(x^4 + A)^2), n))} /* Michael Somos, Sep 29 2006 */

a(n) = A002325(8n+1). [Hirschhorn]
Expansion of q^(-1/8) * eta(q^2)^7 / (eta(q)^3 * eta(q^4)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 29 2006
Euler transform of period 4 sequence [ 3, -4, 3, -2, ...]. - Michael Somos, Sep 29 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139093. - Michael Somos, Mar 16 2011
G.f.: (Sum_{k} x^(k^2)) * (Sum_{k>0} x^((k^2 - k)/2)). - Michael Somos, Sep 29 2006

A112604 Number of representations of n as a sum of three times a square and two times a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Number of representations of 2n as a sum of three times a triangular number and a triangular number.

			a(12) = 3 since we can write 12 = 3(2)^2 + 0 = 3(-2)^2 + 0 = 0 + 2*6.
2*12 = 24 = 3*1+21 = 3*3+15 = 3*6+6 so a(12) = 3.
G.f. = 1 + x^2 + 2*x^3 + 2*x^5 + x^6 + 2*x^9 + 3*x^12 + 2*x^14 + 2*x^15 + ... - _Michael Somos_, Aug 11 2009
G.f. = q + q^9 + 2*q^13 + 2*q^21 + q^25 + 2*q^37 + 3*q^49 + 2*q^57 + 2*q^61 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A112606(n) = a(2*n). 2 * A112607(n) = a(2*n + 1). A123884(n) = a(3*n). A112605(n) = a(3*n + 2). A131961(n) = a(6*n). A112608(n) =a(6*n + 2). 2 * A131963(n) = a(6*n + 3). 2 * A112609(n) = a(6*n + 5). - Michael Somos, Aug 11 2009
Cf. A002324, A033762, A093766, A164272.
Cf. A000700, A000122, A010054, A121373.

a[n_] := DivisorSum[4n+1, Switch[Mod[#, 3], 1, 1, 2, -1, 0, 0]&]; Table[ a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n) = if(n<0, 0, n=4*n+1; sumdiv(n, d, (d%3==1) - (d%3==2)))};

{a(n) = my(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^6+A)^5 / eta(x^2+A)*(eta(x^4+A) / eta(x^3+A) / eta(x^12+A))^2, n))}; /* Michael Somos, Feb 14 2006 */

a(n) = A002324(4n+1) = A033762(2n) = d_{1, 3}(4n+1) - d_{2, 3}(4n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
From Michael Somos, Feb 14 2006: (Start)
Expansion of (psi(q)psi(q^3) + psi(-q)psi(-q^3))/2 in powers of q^2 where psi() is a Ramanujan theta function.
G.f.: (Sum_{k} x^k^2)^3*(Sum_{k>0} x^((k^2-k)/2))^2 = Product_{k>0} (1-x^(4k))(1-x^(6k))(1+x^(2k))(1+x^(3k))^2/(1+x^(6k))^2.
Euler transform of period 12 sequence [0, 1, 2, -1, 0, -2, 0, -1, 2, 1, 0, -2, ...]. (End)
From Michael Somos, Aug 11 2009: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164272.
a(3*n + 1) = 0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 24 2023

A112605 Number of representations of n as a sum of a square and six times a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2 + 6*3.
G.f. = 1 + 2*x + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^15 + 2*x^16 + ... - _Michael Somos_, Aug 11 2009
G.f. = q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A112608(n) = a(2*n). 2 * A112609(n) = a(2*n + 1). A112604(n) = a(3*n). 2 * A121361(n) = a(3*n + 1). A112606(n) = a(6*n). 2 * A131962(n) = a(6*n + 1). 2 * A112607(n) = a(6*n + 3). 2 * A131964(n) = a(6*n + 4). - Michael Somos, Aug 11 2009

a[n_] := DivisorSum[4n+3, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n) = if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))}; /* Michael Somos, May 20 2006 */

{a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2 / eta(x+A)^2 / eta(x^4+A)^2 / eta(x^6+A), n))}; /* Michael Somos, May 20 2006 */

a(n) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-3/4)*eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - Michael Somos, May 20 2006
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - Michael Somos, May 20 2006
a(n)=A002324(4n+3). - Michael Somos, May 20 2006
Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, May 20 2006, Sep 29 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164273. - Michael Somos, Aug 11 2009
a(3*n + 2) = 0. - Michael Somos, Aug 11 2009

A112606 Number of representations of n as a sum of six times a square and a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

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

The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - R. J. Mathar, Apr 28 2020

			1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...
q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...
a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A112604, A112608, A123484, A131961.

a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* Michael Somos, Jun 16 2011 since V6 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* Michael Somos, Jun 16 2011 *)

{a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, d)))} /* Michael Somos, Sep 29 2006 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* Michael Somos, Sep 29 2006 */

a(n) = d_{1, 3}(8n+1) - d_{2, 3}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - Michael Somos, Sep 29 2006
a(n) = A123484(24*n + 3) = A112604(2*n) = A112608(3*n). A131961(n) = a(3*n). A112608(n) = a(3*n + 1).

A112607 Number of representations of n as a sum of a triangular number and twelve times a triangular number.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			a(15) = 2 since we can write 15 = 15 + 12*0 = 3 + 12*1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A123484(24n+15) = 2*a(n). A112609(3n+4) = a(n).

a[n_] := DivisorSum[8n+13, KroneckerSymbol[-3, #]&]/2; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n)=if(n<0, 0, n=8*n+13; sumdiv(n, d, kronecker(-3,d))/2)} /* Michael Somos, Sep 29 2006 */

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^24+A)^2/eta(x+A)/eta(x^12+A), n))} /* Michael Somos, Sep 29 2006 */

a(n) = 1/2*( d_{1, 3}(8n+13) - d_{2, 3}(8n+13) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-13/8)*(eta(q^2)*eta(q^24))^2/(eta(q)*eta(q^12)) in powers of q. - Michael Somos, Sep 29 2006
Expansion of psi(q)*psi(q^12) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 29 2006
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
a(3n+2)=0. - Michael Somos, Sep 29 2006

cp's OEIS Frontend

User: James Sellers

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A385520 Expansion of Product_{k>0} ((1 - q^(2k))(1 - q^(6k))^3)/((1 - q^k)(1 - q^(3k))(1 - q^(4k))(1 - q^(12*k))).