A351007 -id:A351007 - cp's OEIS frontend

A053251 Coefficients of the '3rd-order' mock theta function psi(q).

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 19, 22, 24, 27, 31, 34, 37, 42, 46, 51, 57, 62, 68, 76, 83, 91, 101, 109, 120, 132, 143, 156, 171, 186, 202, 221, 239, 259, 283, 306, 331, 360, 388, 420, 455, 490, 529, 572, 616, 663, 716, 769, 827

Offset: 0

Dean Hickerson, Dec 19 1999

Number of partitions of n into odd parts such that if a number occurs as a part then so do all smaller positive odd numbers.
Number of ways to express n as a partial sum of 1 + [1,3] + [1,5] + [1,7] + [1,9] + .... E.g., a(6)=2 because we have 6 = 1+1+1+1+1+1 = 1+3+1+1. - Jon Perry, Jan 01 2004
Also number of partitions of n such that the largest part occurs exactly once and all the other parts occur exactly twice. Example: a(9)=4 because we have [9], [7,1,1], [5,2,2] and [3,2,2,1,1]. - Emeric Deutsch, Mar 08 2006
Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 < d2/2 < ... < dm/m. - Seiichi Manyama, Mar 17 2018
For Emeric Deutsch's comment above, (1) this appears to be an alternately equal case of A122130, (2) the ordered version (compositions) is A239327, (3) allowing any length gives A351006, (4) the even-length version is A351007. - Gus Wiseman, Feb 25 2022

			q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...
From _Seiichi Manyama_, Mar 17 2018: (Start)
n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m)
--+--------------------------+-------------------------
1 | (1)                      | (1)
2 | (2)                      | (2)
3 | (3)                      | (3)
4 | (4)                      | (4)
  | (1, 3)                   | (1, 3/2)
5 | (5)                      | (5)
  | (1, 4)                   | (1, 2)
6 | (6)                      | (6)
  | (1, 5)                   | (1, 5/2)
7 | (7)                      | (7)
  | (1, 6)                   | (1, 3)
  | (2, 5)                   | (2, 5/2)
8 | (8)                      | (8)
  | (1, 7)                   | (1, 7/2)
  | (2, 6)                   | (2, 3)
9 | (9)                      | (9)
  | (1, 8)                   | (1, 4)
  | (2, 7)                   | (2, 7/2)
  | (1, 3, 5)                | (1, 3/2, 5/3) (End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.13).
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
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, A053252, A053253, A053254, A053255.
Cf. A003475.
Cf. A035363, A035457, A122129, A122130, A122134, A122135, A351003, A351005.

f:=n->q^(n^2)/mul((1-q^(2*i+1)),i=0..n-1); add(f(i),i=1..6);
# second Maple program:
b:= proc(n, i) option remember; (s-> `if`(n>s, 0, `if`(n=s, 1,
      b(n, i-1)+b(n-i, min(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= n-> `if`(n=0, 0, add(b(j, min(j, n-2*j-1)), j=0..iquo(n, 2))):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}]
(* Second program: *)
b[n_, i_] := b[n, i] = Function[s, If[n > s, 0, If[n == s, 1, b[n, i - 1] + b[n - i, Min[n - i, i - 1]]]]][i*(i + 1)/2];
a[n_] := If[n==0, 0, Sum[b[j, Min[j, n-2*j-1]], {j, 0, Quotient[n, 2]}]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 17 2018, after Alois P. Heinz *)

{ n=20; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]+2*i-1)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

{a(n) = local(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k-1) / (1 - x^(2*k-1)) + O(x^(n-(k-1)^2+1))), n))} /* Michael Somos, Sep 04 2007 */

G.f.: psi(q) = Sum_{n>=1} q^(n^2) / ( (1-q)*(1-q^3)*...*(1-q^(2*n-1)) ).
G.f.: Sum_{k>=1} q^k*Product_{j=1..k-1} (1+q^(2*j)) (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006
a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 08 2006

A351005 Number of integer partitions of n into parts that are alternately equal and unequal.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 2, 4, 2, 5, 4, 6, 4, 8, 5, 10, 6, 12, 8, 16, 9, 18, 12, 22, 14, 28, 16, 33, 20, 40, 24, 48, 28, 56, 34, 67, 40, 80, 46, 94, 56, 110, 64, 130, 75, 152, 88, 176, 102, 206, 118, 238, 138, 276, 159, 320, 182, 368, 210, 424, 242, 488, 276, 558

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

Also partitions whose multiplicities are all 2's, except possibly for the last, which may be 1.

			The a(1) = 1 through a(12) = 6 partitions (A..C = 10..12):
  1  2   3  4   5    6     7    8     9    A     B      C
     11     22  221  33    331  44    441  55    443    66
                     2211       332        442   551    552
                                3311       3322  33221  4422
                                           4411         5511
                                                        332211

The even-length ordered version is A003242, ranked by A351010.
The even-length case is A035457.
Without equalities we have A122135, opposite A122129, even-length A122134.
The non-strict version is A351004, opposite A351003, even-length A035363.
The opposite version is A351006, even-length A351007.
Cf. A000070, A018819, A027383, A088218, A350842, A350844, A351008.

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

A351006 Number of integer partitions of n into parts that are alternately unequal and equal.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 6, 7, 9, 10, 12, 14, 16, 18, 22, 25, 28, 31, 36, 40, 46, 50, 56, 64, 71, 78, 88, 96, 106, 118, 130, 143, 158, 172, 190, 209, 228, 248, 274, 298, 324, 354, 384, 418, 458, 494, 536, 584, 631, 683, 742, 800, 864, 936, 1010, 1088, 1176, 1264

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

			The a(1) = 1 through a(11) = 12 partitions (A = 10, B = 11):
  1  2  3   4    5    6    7    8     9      A      B
        21  31   32   42   43   53    54     64     65
            211  41   51   52   62    63     73     74
                 311  411  61   71    72     82     83
                           322  422   81     91     92
                           511  611   522    433    A1
                                3221  711    622    533
                                      4221   811    722
                                      32211  5221   911
                                             42211  4331
                                                    6221
                                                    52211

Without equalities we have A122129, opposite A122135, even-length A351008.
The non-strict version is A351003, opposite A351004, even-length A351012.
The alternately equal and unequal version is A351005, even-length A035457.
The even-length case is A351007.
Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A344605, A350842, A350844, A351011.

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

A122129 Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 30, 37, 46, 57, 69, 84, 102, 123, 148, 177, 211, 252, 299, 353, 417, 491, 576, 675, 789, 920, 1071, 1244, 1442, 1670, 1929, 2224, 2562, 2946, 3381, 3876, 4437, 5072, 5791, 6602, 7517, 8551, 9714, 11021, 12493, 14145

Offset: 0

Michael Somos, Aug 21 2006

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
a(n) = number of SE partitions of n, for n >= 1; see A237981. - Clark Kimberling, Mar 19 2014
In Watson 1937 page 275 he writes "Psi_0(1,q) = prod_1^oo (1+q^{2n}) G(q^8)" so this is the expansion in powers of q^2. - Michael Somos, Jun 28 2015
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(x) (see A003114), H(x) (A003106).
From Gus Wiseman, Feb 19 2022: (Start)
This appears to be the number of integer partitions of n with every other pair of adjacent parts strictly decreasing, as in the pattern a > b >= c > d >= e for a partition (a, b, c, d, e). For example, the a(1) = 1 through a(9) = 12 partitions are:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)    (8)     (9)
            (21)  (31)   (32)   (42)   (43)   (53)    (54)
                  (211)  (41)   (51)   (52)   (62)    (63)
                         (311)  (321)  (61)   (71)    (72)
                                (411)  (322)  (422)   (81)
                                       (421)  (431)   (432)
                                       (511)  (521)   (522)
                                              (611)   (531)
                                              (3221)  (621)
                                                      (711)
                                                      (4221)
                                                      (32211)
The even-length case is A351008. The odd-length case appears to be A122130. Swapping strictly and weakly decreasing relations appears to give A122135. The alternately unequal and equal case is A351006, strict A035457, opposite A351005, even-length A351007. (End)
Wiseman's first conjecture above was proved by Gordon, Theorem 7. For two other combinatorial interpretations of this sequence see Connor, Proposition 1. - Peter Bala, Dec 22 2024

			Clark Kimberling's SE partition comment, n=6: the 5 SE partitions are [1,1,1,1,1,1] from the partitions 6 and 1^6; [1,1,1,2,1] from 5,1 and 2,1^4; [1,1,3,1] from 4,2 and 2^2,1^2; [2,3,1] from 3,2,1 and 3^2 and 2^3; and [1,2,2,1] from 4,1^2 and 3,1^3. - _Wolfdieter Lang_, Mar 20 2014
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 9*x^8 + ...
G.f. = 1/q + q^39 + q^79 + 2*q^119 + 3*q^159 + 4*q^199 + 5*q^239 + ...

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.7). MR0858826 (88b:11063)
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(a), p. 591.

Alois P. Heinz, 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.
Basil Gordon, Continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741-748. MR 32 # 1477.
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 1. [From _N. J. A. Sloane_, Mar 19 2012]
Michael Somos, Introduction to Ramanujan theta functions
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A000070, A096441, A122134, A351004.

f:=n->1/mul(1-q^(20*k+n),k=0..20);
f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19);
series(%,q,200); seriestolist(%); # N. J. A. Sloane, Mar 19 2012.
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, 1, 0,
       1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1]
      [1+irem(d, 20)], d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 12 2013

a[0] = 1; a[n_] := a[n] = Sum[Sum[d*{0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1}[[1+Mod[d, 20]]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jan 10 2014, after Alois P. Heinz *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ x, x, 2 k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^4, x^20] QPochhammer[ x^16, x^20]), {x, 0, n}]; (* Michael Somos, Jun 28 2015 *)

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

Euler transform of period 20 sequence [ 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, ...].
Expansion of f(-x^2) * f(-x^20) / (f(-x) * f(-x^4,-x^16)) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(x^3, x^7) / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 28 2015
Expansion of f(-x^8, -x^12) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jun 28 2015
Expansion of G(x^4) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 28 2015
G.f.: Sum_{k>=0} x^k^2 / ((1 - x) * (1 - x^2) ... (1 - x^(2*k))).
G.f.: 1 / (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(20*k-4)) * (1 - x^(20*k-16))).
Let f(n) = 1/Product_{k >= 0} (1 - q^(20k+n)). Then g.f. is f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19). - N. J. A. Sloane, Mar 19 2012
a(n) is the number of partitions of n into parts that are either odd or == +-4 (mod 20). - Michael Somos, Jun 28 2015
a(n) ~ (3+sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A122135 Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 26, 31, 40, 48, 60, 72, 89, 106, 130, 154, 186, 220, 264, 310, 370, 433, 512, 598, 704, 818, 958, 1110, 1293, 1494, 1734, 1996, 2308, 2650, 3052, 3496, 4014, 4584, 5248, 5980, 6825, 7760, 8834, 10020, 11380, 12882, 14594

Offset: 0

Michael Somos, Aug 21 2006

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
From Gus Wiseman, Feb 26 2022: (Start)
Conjecture: Also the number of integer partitions y of n such that y_i > y_{i+1} for all even i. For example, the a(1) = 1 through a(9) = 12 partitions are:
  (1)  (2)   (3)   (4)   (5)    (6)     (7)     (8)     (9)
       (11)  (21)  (22)  (32)   (33)    (43)    (44)    (54)
                   (31)  (41)   (42)    (52)    (53)    (63)
                         (221)  (51)    (61)    (62)    (72)
                                (321)   (331)   (71)    (81)
                                (2211)  (421)   (332)   (432)
                                        (3211)  (431)   (441)
                                                (521)   (531)
                                                (3311)  (621)
                                                (4211)  (3321)
                                                        (4311)
                                                        (5211)
The even-length case appears to be A122134.
The odd-length case is A351595.
The alternately unequal version appears to be A122129, even A351008, odd A122130.
The alternately equal version is A351003, even A351012, odd A000009.
The alternately equal and unequal version is A351005, even A035457, odd A351593.
The alternately unequal and equal version is A351006, even A351007, odd A053251. (End)
For Wiseman's conjecture above and three other partition-theoretic interpretations of this sequence see Connor, Proposition 4. - Peter Bala, Jan 02 2025

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
G.f. = q^9 + q^49 + 2*q^89 + 2*q^129 + 3*q^169 + 4*q^209 + 6*q^249 + ...

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.5). MR0858826 (88b:11063)
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(d), p. 591.

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.
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 2. [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], Sep 30 2015.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A003242, A035363, A053251, A122129, A122130, A122134.

f:=n->1/mul(1-q^(20*k+n),k=0..20);
f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19);
series(%,q,200); seriestolist(%); # N. J. A. Sloane, Mar 19 2012

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^5] QPochhammer[ x^4, -x^5] QPochhammer[-x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 12 2016 *)
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+1))*(1 - x^(20*k+2))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+8))*(1 - x^(20*k+9))*(1 - x^(20*k+11))*(1 - x^(20*k+12))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+18))*(1 - x^(20*k+19)) ), {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, 1 - x^i, 1 + x * O(x^(n-k^2-k)))), n))};

Expansion of f(x^2, x^8) / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 12 2016
Expansion of f(-x^3, -x^7) * f(-x^4, -x^16) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1))).
Let f(n) = 1/Product_{k >= 0} (1-q^(20k+n)). Then g.f. is f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19); - N. J. A. Sloane, Mar 19 2012.
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

A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The ordered version (compositions) is A027383.
The version for unequal instead of equal is A122135, even-length A351008.
For odd instead of even indices we have A351004, even-length A035363.
Requiring inequalities at odd positions gives A351006, even-length A351007.
The even-length case is A351012.
Cf. A000070, A018819, A088218, A101417, A122129, A350837, A351005.

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

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}]

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

cp's OEIS Frontend

A053251 Coefficients of the '3rd-order' mock theta function psi(q).

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A351005 Number of integer partitions of n into parts that are alternately equal and unequal.

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A351006 Number of integer partitions of n into parts that are alternately unequal and equal.

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A122129 Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).

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A122135 Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.

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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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