A333374 -id:A333374 - cp's OEIS frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164

Offset: 0

N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

Also number of partitions of n where no part appears more than three times.
a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).
Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006
Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010
Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010
The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015
Convolution inverse is A082303. - Michael Somos, Sep 30 2017
The g.f. in the form Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1-x^k) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1+x^k-2*x^k) == Sum_{n >= 0} x^(n*(n+1)/2) (mod 2). It follows that a(n) is odd iff n = k*(k + 1)/2 for some nonnegative integer k. Cf. A333374. - Peter Bala, Jan 08 2025

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...
a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).
M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See Theorem 1.4 p. 75.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Cristina Ballantine and Mircea Merca, Parity of sums of partition numbers and squares in arithmetic progressions, The Ramanujan Journal, 2016.
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129.]
S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.
Eric Weisstein's World of Mathematics, Partition Function b_k and Partition Function P.
Wikipedia, Glaisher's Theorem.

Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601.
Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010
Cf. A174715. - Gary W. Adamson, Mar 27 2010
Cf. A082303.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a001935 = p a042968_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Sep 02 2012

g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..48); # Emeric Deutsch, Apr 19 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j,1,48}], {x,0,48}],x] (* Jean-François Alcover, May 26 2011, after Jon Perry *)
QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* Robert Price, Jul 28 2020 *)

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

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* Michael Somos, Jun 01 2004 */

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)),n)} \\ Paul D. Hanna, Jul 24 2013

Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].
Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - Michael Somos, Mar 19 2004
Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011
G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004
G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - Jon Perry, May 05 2005
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).
G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011
G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - Joerg Arndt, Jun 21 2011
A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).
G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013
a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - Michael Somos, Sep 30 2017

More terms from James Sellers

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 55, 90, 144, 226, 346, 522, 777, 1138, 1648, 2362, 3348, 4704, 6554, 9056, 12425, 16932, 22922, 30848, 41282, 54946, 72768, 95914, 125842, 164402, 213901, 277204, 357904, 460448, 590330, 754368, 960948, 1220370, 1545306

Offset: 0

N. J. A. Sloane

The Cayley reference is actually to A079006. - Michael Somos, Feb 24 2011
In the math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of 4-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The g.f. in the form A(x) = Sum_{k >= 0} x^(k*(k+1)) / (1 + 2*Sum_{k >= 1} (-1)^k * x^(k^2)) == Sum_{k >= 0} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ...
G.f. = q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..1000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
joro on Math Overflow, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?.
T. Kathiravan and S. N. Fathima, On L-regular bipartitions modulo L, The Ramanujan Journal 44.3 (2017): 549-558.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A022568, A079006, A082304, A098613, A127391, A127392, A210656.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548, A333374.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> [2,2,2,0] [modp(n-1,4)+1]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
f:=(k,M) -> mul(1-q^(k*j),j=1..M); LRBP := (L,M) -> (f(L,M)/f(1,M))^2; S := L -> seriestolist(series(LRBP(L,80),q,60)); S(4); # N. J. A. Sloane, Oct 20 2019

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-François Alcover, Sep 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

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

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))};

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
Convolution square of A001935. A079006(n) = (-1)^n a(n).
Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].
Given g.f. A(x), then B(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^2.
Equals A000009 convolved with A098613. - Gary W. Adamson, Mar 24 2011
a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012
Convolution inverse is A082304. - Michael Somos, May 16 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015
Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015
a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 18 2015
G.f.: A(x) = Sum_{n >= 0} x^(n*(n+1)) / Sum_{n = -oo..oo} (-1)^n*x^(n^2). - Peter Bala, Feb 19 2021

A066447 Number of basis partitions (or basic partitions) of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 10, 13, 16, 20, 26, 32, 40, 50, 61, 74, 90, 108, 130, 156, 186, 222, 264, 313, 370, 436, 512, 600, 702, 818, 952, 1106, 1282, 1484, 1715, 1978, 2278, 2620, 3008, 3448, 3948, 4512, 5150, 5872, 6684, 7600, 8632, 9791, 11094, 12558, 14198, 16036, 18096, 20398

Offset: 0

Herbert S. Wilf, Dec 29 2001

The k-th successive rank of a partition pi = (pi_1, pi_2, ..., pi_s) of the integer n is r_k = pi_k - pi'_k, where pi' denotes the conjugate partition. A partition pi is basic if the number of dots in its Ferrers diagram is the least among all the Ferrers diagrams of partitions with the same rank vector.

The g.f. Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) = Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{n >= 0} x^(n^2) (mod 2). It follows that a(n) is odd iff n is a square (Nolan et al., equation 6, p. 282). - Peter Bala, Jan 08 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
J. M. Nolan, C. D. Savage and H. S. Wilf, Basis partitions, Discrete Math. 179 (1998), 277-283.

Row sums of A066448.
Cf. A001130, A001935, A003114, A306734, A333374.
Cf. A192918, A376841.

b := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(b(n-d,d)+b(n-2*d+1,d-1)+b(n-3*d+1,d-1)) fi: end: A066447 := n->add(b(n,d),d=0..n);

nmax = 60; CoefficientList[Series[Sum[x^(n^2)*Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 17 2020 *)
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k - 1), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 17 2020 *)

N=66; x='x+O('x^N); s=sum(n=0,N,x^(n^2)*prod(k=1,n,(1+x^k)/(1-x^k))); Vec(s) /* Joerg Arndt, Apr 07 2011 */

G.f.: Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) [Given in Nolan et al. reference]. - Joerg Arndt, Apr 07 2011
Limit_{n->infinity} a(n) / A333374(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... - Vaclav Kotesovec, Mar 17 2020
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.193340468476900308848561788251945... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 + cosh(arccosh(53*sqrt(11/2)/64)/3) / (3*sqrt(22))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 19 2020, updated Oct 10 2024

cp's OEIS Frontend

A376841 Decimal expansion of a constant related to the asymptotics of A066447 and A333374.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A066447 Number of basis partitions (or basic partitions) of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A376852 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A376852 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.