A035457 -id:A035457 - cp's OEIS frontend

A122134 Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 18, 24, 28, 36, 42, 54, 62, 78, 91, 112, 130, 159, 184, 222, 258, 308, 356, 424, 488, 576, 664, 778, 894, 1044, 1196, 1389, 1590, 1838, 2098, 2419, 2754, 3162, 3596, 4114, 4668, 5328, 6032, 6864, 7760, 8806, 9936, 11252

Offset: 0

Michael Somos, Aug 21 2006, Oct 10 2007

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
In Watson 1937 page 275 he writes "Psi_0(q^{1/2},q) = prod_1^oo (1+q^{2n}) G(-q^2)" so this is the expansion in powers of q^2. - Michael Somos, Jun 29 2015
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
From Gus Wiseman, Feb 26 2022: (Start)
Conjecture: Also the number of even-length integer partitions y of n such that y_i != y_{i+1} for all even i. For example, the a(2) = 1 through a(9) = 7 partitions are:
  (11)  (21)  (22)  (32)  (33)    (43)    (44)    (54)
              (31)  (41)  (42)    (52)    (53)    (63)
                          (51)    (61)    (62)    (72)
                          (2211)  (3211)  (71)    (81)
                                          (3311)  (3321)
                                          (4211)  (4311)
                                                  (5211)
This appears to be the even-length version of A122135.
The odd-length version is A351595.
Cf. A035363, A035457, A350842, A350844, A351003-A351012, A351593. (End)
For Wiseman's conjecture above and three other partition-theoretic interpretations of this sequence see Connor, Proposition 3. - Peter Bala, Jan 02 2025

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = q + q^81 + q^121 + 2*q^161 + 2*q^201 + 4*q^241 + 4*q^281 + ...

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(c), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.6). MR0858826 (88b:11063)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Willard G. Connor, Partition Theorems Related to Some Identities of Rogers and Watson, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
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.
Mircea Merca, From a Rogers's identity to overpartitions, Periodica Mathematica Hungarica, Vol. 75, issue 2, 172-179, 2017.
Michael Somos, Introduction to Ramanujan theta functions
G. N. Watson, The Mock Theta Functions (2), Proceedings of the London Mathematical Society, s2-42: 274-304, 1937.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A053251, A122129, A122130, A122135.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / QPochhammer[ x, x, 2 k], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jun 29 2015 *)
a[ n_] := SeriesCoefficient [ 1 / (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2]), {x, 0, n}]; (* Michael Somos, Jun 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, -x^5] QPochhammer[ -x^3, -x^5] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 29 2015 *)
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+2))*(1 - x^(20*k+3))*(1 - x^(20*k+4))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+7))*(1 - x^(20*k+13))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+16))*(1 - x^(20*k+17)) *(1 - x^(20*k+18))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)

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

Euler transform of period 20 sequence [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, ...].
Expansion of f(x^4, x^6) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 29 2015
Expansion of f(-x^2, x^3) / phi(-x^2) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jun 29 2015
Expansion of G(-x) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 29 2015
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))).
Expansion of f(-x, -x^9) * f(-x^8, -x^12) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is the Ramanujan general theta function.
a(n) = number of partitions of n into parts that are each either == 2, 3, ..., 7 (mod 20) or == 13, 14, ..., 18 (mod 20). - Michael Somos, Jun 29 2015 [corrected by Vaclav Kotesovec, Nov 12 2016]
a(n) ~ (3 - sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016

A357643 Number of integer compositions of n into parts that are alternately equal and unequal.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 5, 5, 9, 7, 17, 14, 28, 25, 49, 42, 87, 75, 150, 132, 266, 226, 466, 399, 810, 704, 1421, 1223, 2488, 2143, 4352, 3759, 7621, 6564, 13339, 11495, 23339, 20135, 40852, 35215, 71512, 61639, 125148, 107912, 219040, 188839, 383391, 330515, 670998

Offset: 0

Gus Wiseman, Oct 12 2022

nonn

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)  (4)    (5)    (6)     (7)      (8)
       (11)       (22)   (113)  (33)    (115)    (44)
                  (112)  (221)  (114)   (223)    (116)
                                (1122)  (331)    (224)
                                (2211)  (11221)  (332)
                                                 (1133)
                                                 (3311)
                                                 (22112)
                                                 (112211)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The even-length version is A003242, ranked by A351010, partitions A035457.
Without equal relations we have A016116, equal only A001590 (apparently).
The version for partitions is A351005.
The opposite version is A357644, partitions A351006.
A011782 counts compositions.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357645 counts compositions by half-alternating sum, skew A357646.
Cf. A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,15}]

C_x(N) = {my(x='x+O('x^N), h=(1+sum(k=1,N, (x^k)/(1+x^(2*k))))/(1-sum(k=1,N, (x^(2*k))/(1+x^(2*k))))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, May 28 2024

G.f.: (1 + Sum_{k>0} (x^k)/(1 + x^(2*k)))/(1 - Sum_{k>0} (x^(2*k))/(1 + x^(2*k))). - John Tyler Rascoe, May 28 2024

More terms from Alois P. Heinz, Oct 12 2022

A357644 Number of integer compositions of n into parts that are alternately unequal and equal.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 8, 13, 17, 25, 30, 44, 58, 77, 98, 142, 176, 245, 311, 426, 548, 758, 952, 1319, 1682, 2308, 2934, 4059, 5132, 7087, 9008, 12395, 15757, 21728, 27552, 38019, 48272, 66515, 84462, 116467, 147812, 203825, 258772, 356686, 452876, 624399, 792578

Offset: 0

Gus Wiseman, Oct 14 2022

nonn

			The a(1) = 1 through a(7) = 13 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (23)   (24)    (25)
                  (211)  (32)   (42)    (34)
                         (41)   (51)    (43)
                         (122)  (411)   (52)
                         (311)  (1221)  (61)
                                (2112)  (133)
                                        (322)
                                        (511)
                                        (2113)
                                        (3112)
                                        (12211)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Without equal relations we have A000213, equal only A027383.
Even-length opposite: A003242, ranked by A351010, partitions A035457.
The version for partitions is A351006.
The opposite version is A357643, partitions A351005.
A011782 counts compositions.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357645 counts compositions by half-alternating sum, skew A357646.
Cf. A001590, A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,10}]

More terms from Alois P. Heinz, Oct 19 2022

A053253 Coefficients of the '3rd-order' mock theta function omega(q).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, 36, 44, 56, 68, 82, 101, 122, 146, 176, 210, 248, 296, 350, 410, 484, 566, 660, 772, 896, 1038, 1204, 1391, 1602, 1846, 2120, 2428, 2784, 3182, 3628, 4138, 4708, 5347, 6072, 6880, 7784, 8804, 9940, 11208, 12630

Offset: 0

Dean Hickerson, Dec 19 1999

Empirical: a(n) is the number of integer partitions mu of 2n+1 such that the diagram of mu has an odd number of cells in each row and in each column. - John M. Campbell, Apr 24 2020
From Gus Wiseman, Jun 26 2022: (Start)
By Campbell's conjecture above that a(n) is the number of partitions of 2n+1 with all odd parts and all odd conjugate parts, the a(0) = 1 through a(5) = 8 partitions are (B = 11):
  (1)  (3)    (5)      (7)        (9)          (B)
       (111)  (311)    (511)      (333)        (533)
              (11111)  (31111)    (711)        (911)
                       (1111111)  (51111)      (33311)
                                  (3111111)    (71111)
                                  (111111111)  (5111111)
                                               (311111111)
                                               (11111111111)
These partitions are ranked by A352143. (End)

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Steven Charlton, Explicit linear dependence congruence relations for the partition function modulo 4, arXiv:2412.17459 [math.NT], 2024. See p. 3.
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053254, A053255, A261401.
Cf. A095913(n)=a(n-3).
Cf. A259094.
Conjectured to count the partitions ranked by A352143.
A069911 = strict partitions w/ all odd parts, ranked by A258116.
A078408 = partitions w/ all odd parts, ranked by A066208.
A117958 = partitions w/ all odd parts and multiplicities, ranked by A352142.
Cf. A000009, A000290, A000701, A035363 (complement A086543), A035444, A035457, A045931, A055922, A258117.

Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}]^2, {n, 0, 6}], {q, 0, 100}]

{a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); polcoeff( sum(k=0, (sqrtint(2*n+1)-1)\2, A*=(x^(4*k)/(1-x^(2*k+1))^2 +x*O(x^(n-2*(k^2-k))))), n))} /* Michael Somos, Aug 18 2006 */

{a(n)=local(A); if(n<0, 0, n++; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* Michael Somos, Aug 18 2006 */

G.f.: omega(q) = Sum_{n>=0} q^(2*n*(n+1))/((1-q)*(1-q^3)*...*(1-q^(2*n+1)))^2.
G.f.: Sum_{k>=0} x^k/((1-x)(1-x^3)...(1-x^(2k+1))). - Michael Somos, Aug 18 2006
G.f.: (1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 10 2019
Conjectural g.f.: 1/(1 - x)*( 1 + Sum_{n >= 0} x^(3*n+1) /((1 - x)*(1 - x^3)*...*(1 - x^(2*n+1))) ). - Peter Bala, Nov 18 2024

A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.

			The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
  21  31  32  42  43  53    54    64    65    75    76    86    87
          41  51  52  62    63    73    74    84    85    95    96
                  61  71    72    82    83    93    94    A4    A5
                      3221  81    91    92    A2    A3    B3    B4
                            4221  5221  A1    B1    B2    C2    C3
                                        4331  4332  C1    D1    D2
                                        6221  5331  5332  5441  E1
                                              7221  6331  6332  5442
                                                    8221  7331  6441
                                                          9221  7332
                                                                8331
                                                                A221
                                                                433221

The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A351006, odd-length A053251.
Without equalities we have A351008, any length A122129, opposite A122135.
Without inequalities we have A351012, any length A351003, opposite A351004.
Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 23, 26, 33, 37, 46, 52, 63, 72, 87, 98, 117, 133, 157, 178, 209, 236, 276, 312, 361, 408, 471, 530, 609, 686, 784, 881, 1004, 1126, 1279, 1433, 1621, 1814, 2048, 2286, 2574, 2871, 3223, 3590, 4022, 4472, 5000

Offset: 0

N. J. A. Sloane, May 05 2002

Number 39 of the 130 identities listed in Slater 1952.

Number of partitions of 2*n into distinct odd parts. - Vladeta Jovovic, May 08 2003

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + ...
G.f. = q^-1 + q^95 + q^143 + 2*q^191 + 2*q^239 + 3*q^287 + 3*q^335 + ...

M. D. Hirschhorn, The Power of q, Springer, 2017. Chapter 19, Exercises p. 173.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities, Quaestiones Math., 24 (2001), 403-416.
George E. Andrews, Jethro van Ekeren and Reimundo Heluani, The singular support of the Ising model, arXiv:2005.10769 [math.QA], 2020. See (1.4.2) p. 2.
T. Gannon, G. Hoehn, H. Yamauchi, et. al., VOA Unitary Minimal Model m=1, character.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 4. [From _N. J. A. Sloane_, Mar 19 2012]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp. 147-167, (1952).
Eric Weisstein's World of Mathematics, Jackson-Slater Identity
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - _N. J. A. Sloane_, Dec 25 2012

Cf. A000700, A027356, A035457, A069908, A069909, A069911, A122129, A122135.

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0$2, 1$4, 0$5, 1$4, 0][irem(d, 16)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

max = 56; p = Product[1/(1-x^i), {i, Select[Range[max], MemberQ[{2, 3, 4, 5, 11, 12, 13, 14}, Mod[#, 16]]&]}]; s = Series[p, {x, 0, max}]; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Apr 09 2014 *)
nmax=60; CoefficientList[Series[Product[(1-x^(8*k-1))*(1-x^(8*k-7))*(1-x^(8*k))*(1-x^(16*k-6))*(1-x^(16*k-10))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0 }[[ Mod[k, 16] + 1]], {k, n}], {x, 0, n}]; (* Michael Somos, Apr 14 2016 *)

{a(n) = my(A); if( n<0,0, n=2*n; A = x * O(x^n); polcoeff( eta(-x + A) / eta(x^2 + A), n))}; /* Michael Somos, Apr 11 2004 */

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=0, S, q^(2*n^2) / prod(k=1, 2*n, 1-q^k ) );
Vec(gf)  \\ Joerg Arndt, Apr 01 2014

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0][k%16 + 1]), n))}; /* Michael Somos, Apr 14 2016 */

Euler transform of period 16 sequence [0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: Sum_{n>=0} q^(2*n^2) / Product_{k=1..2*n} (1 - q^k). - Joerg Arndt, Apr 01 2014
a(n) ~ exp(sqrt(n/3)*Pi) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 04 2015
Expansion of f(x^3, x^5) / f(-x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Apr 14 2016
a(n) = A000700(2*n).
a(n) = A027356(4n+1,2n+1). - Alois P. Heinz, Oct 28 2019
From Peter Bala, Feb 08 2021: (Start)
G.f.: A(x) = Product_{n >= 1} (1 + x^(4*n))^2*(1 + x^(4*n-2))*(1 + x^(8*n-3))*(1 + x^(8*n-5)).
The 2 X 2 matrix Product_{k >= 0} [1, x^(2*k+1); x^(2*k+1), 1] = [A(x^2), x*B(x^2); x*B(x)^2, A(x^2)], where B(x) is the g.f. of A069911.
A(x^2) + x*B(x^2) = A^2(-x) + x*B^2(-x) = Product_{k >= 0} 1 + x^(2*k+1), the g.f. of A000700.
A^2(x) + x*B^2(x) is the g.f. of A226622.
(A^2(x) + x*B^2(x))/(A^2(x) - x*B^2(x)) is the g.f. of A208850.
A^4(sqrt(x)) - x*B^4(sqrt(x)) is the g.f. of A029552.
A(x)*B(x) is the g.f. of A226635; A(-x)/B(-x) is the g.f. of A111374; B(-x)/A(-x) is the g.f. of A092869. (End)

A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

			The a(1) = 1 through a(9) = 7 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    221    33      331      44        333
              1111  11111  222     22111    332       441
                           2211    1111111  2222      22221
                           111111           3311      33111
                                            221111    2211111
                                            11111111  111111111

The ordered version (compositions) is A016116.
The even-length case is A035363.
A reverse version is A096441, both A349060.
The version for unequal instead of equal is A122129, even-length A351008.
The version for even instead of odd indices is A351003, even-length A351012.
The strict version is A351005, opposite A351006, even-length A035457.
Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A069911 Expansion of Product_{i in A069909} 1/(1 - x^i).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 20, 25, 29, 35, 41, 49, 57, 68, 78, 93, 107, 125, 144, 168, 192, 223, 255, 294, 335, 385, 437, 501, 568, 647, 732, 833, 939, 1065, 1199, 1355, 1523, 1717, 1925, 2166, 2425, 2720, 3040, 3405, 3797, 4244, 4727, 5272

Offset: 0

N. J. A. Sloane, May 05 2002

nonn

Arises from an identity of Slater's.
Number of partitions of 2*n+1 into distinct odd parts. - Vladeta Jovovic, May 08 2003
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of partitions of 2n+1 such that if k is the largest part, then k occurs an odd number of times and each integer from 1 to k-1 occurs a positive even number of times. Example: a(4)=2 because we have [3,2,2,1,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 16 2006
Difference between number of partitions of 2n+1 with an odd number of parts and those with an even number of parts (this is a consequence of Jovovic's comment above). - George Beck, May 22 2016
Let b(k) be the convolution inverse of A035457, k=1, 2, 3, ...; then a(n) = -b(4n+3), n = 0, 1, 2, 3, ... (conjectured). - George Beck, Aug 19 2017

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + ...
G.f. = q^23 + q^71 + q^119 + q^167 + 2*q^215 + 2*q^263 + 3*q^311 + 4*q^359 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities Quaestiones Math., 24 (2001), 403-416.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33--37. MR0541341 (80j:05010). See Theorem 3. [From _N. J. A. Sloane_, Mar 19 2012]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

Cf. A000700, A035457, A069908, A069909, A069910, A081362, A122129, A122135.

h:=product(1+x^(2*i-1),i=1..60): hser:=series(h,x=0,120): seq(coeff(hser,x^(2*n+1)),n=0..56); # Emeric Deutsch, Apr 16 2006

H[x_] := x*QPochhammer[-1/x, x^2]/(1 + x); s = (H[Sqrt[x]] - H[-Sqrt[x]]) / (2*Sqrt[x]) + O[x]^60; CoefficientList[s, x] (* Jean-François Alcover, Nov 14 2015, after Emeric Deutsch *)

{a(n) = my(A); if( n<0, 0, n=2*n+1; A = x * O(x^n); -polcoeff( eta(x + A) / eta(x^2 + A), n))}; /* Michael Somos, Jul 18 2006 */

a(n) = A000700(2n+1) = -A081362(2n+1).
Euler transform of period 16 sequence [ 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: ( H(sqrt(x)) - H(-sqrt(x)) ) / (2*sqrt(x)), where H(x)=prod(i>=1, 1+x^(2*i-1) ). - Emeric Deutsch, Apr 16 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 14 2015
Expansion of f(x, x^7) / f(-x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 04 2016

A351008 Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093

Offset: 0

Gus Wiseman, Jan 31 2022

Write the series in the g.f. given below as Sum_{k >= 0} q^(1 + 3 + 5 + ... + 2*k-1 + 2*k)/Product_{i = 1..2*k} 1 - q^i. Since 1/Product_{i = 1..2*k} 1 - q^i is the g.f. for partitions with parts <= 2*k, we see that the k-th summand of the series is the g.f. for partitions with largest part 2*k in which every odd number less than 2*k appears at least once as a part. The partitions of this type are conjugate to (and hence equinumerous with) the partitions (y_1, y_2, ..., y_{2*k}) of even length 2*k having strict decrease y_i > y_(i+1) for all odd i < 2*k. - Peter Bala, Jan 02 2024

			The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):
  21   31   32   42   43   53     54     64     65     75     76
            41   51   52   62     63     73     74     84     85
                      61   71     72     82     83     93     94
                           3221   81     91     92     A2     A3
                                  4221   4321   A1     B1     B2
                                         5221   4331   4332   C1
                                                5321   5331   5332
                                                6221   5421   5431
                                                       6321   6331
                                                       7221   6421
                                                              7321
                                                              8221
a(10) = 6: the six partitions 64, 73, 82, 91, 4321 and 5221 listed above have conjugate partitions 222211, 2221111, 22111111, 211111111, 4321 and 43111, These are the partitions of 10 with largest part L even and such that every odd number less than L appears at least once as a part. - _Peter Bala_, Jan 02 2024

The version for equal instead of unequal is A035363.
The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A122129, opposite A122135.
The odd-length version appears to be A122130.
The alternately unequal and equal version is A351007, any length A351006.
Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.

series(add(q^(n*(n+2))/mul(1 - q^k, k = 1..2*n), n = 0..10), q, 141):
seq(coeftayl(%, q = 0, n), n = 0..140); # Peter Bala, Jan 03 2025

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

Conjecture: a(n+1) = A122129(n+1) - A122130(n). - Gus Wiseman, Feb 21 2022

G.f.: Sum_{n >= 0} q^(n*(n+2))/Product_{k = 1..2*n} 1 - q^k = 1 + q^3 + q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 4*q^8 + 5*q^9 + 6*q^10 + .... - Peter Bala, Jan 02 2024

A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 9, 10, 13, 16, 21, 24, 29, 35, 43, 50, 60, 70, 83, 97, 113, 132, 156, 178, 206, 239, 275, 316, 365, 416, 477, 545, 620, 706, 806, 912, 1034, 1173, 1326, 1496, 1691, 1902, 2141, 2410, 2704, 3034, 3406, 3808, 4261, 4765, 5317, 5932, 6617

Offset: 0

Gus Wiseman, Feb 03 2022

nonn

			The a(2) = 1 through a(8) = 9 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (3111)    (2221)    (71)
                              (111111)  (4111)    (2222)
                                        (211111)  (3221)
                                                  (5111)
                                                  (311111)
                                                  (11111111)

The ordered version (compositions) is A027383(n-2).
For odd instead of even indices we have A035363, any length A351004.
The version for unequal parts appears to be A122134, any length A122135.
This is the even-length case of A351003.
Requiring inequalities at odd positions gives A351007, any length A351006.
Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

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A122134 Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).

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A357643 Number of integer compositions of n into parts that are alternately equal and unequal.

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A357644 Number of integer compositions of n into parts that are alternately unequal and equal.

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A053253 Coefficients of the '3rd-order' mock theta function omega(q).

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A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.

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A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

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A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

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