A035294 -id:A035294 - cp's OEIS frontend

A079122 Number of ways to partition 2*n into distinct positive integers not greater than n.

1, 0, 0, 1, 1, 3, 5, 8, 13, 21, 31, 46, 67, 95, 134, 186, 253, 343, 461, 611, 806, 1055, 1369, 1768, 2270, 2896, 3678, 4649, 5847, 7325, 9141, 11359, 14069, 17367, 21363, 26202, 32042, 39068, 47512, 57632, 69728, 84167, 101365, 121801, 146053, 174777

Offset: 0

Reinhard Zumkeller, Dec 27 2002

nonn

			a(4)=1 [1+3+4=2*4]; a(5)=3 [1+2+3+4=1+4+5=2+3+5=2*5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..80 from Reinhard Zumkeller)

Cf. A035294, A079126, A000009, A079124, A079125, A067953, A111133.

a079122 n = p [1..n] (2 * n) where
   p _  0     = 1
   p [] _     = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Mar 16 2012

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) + `if`(i>n, 0, b(n-i, i-1))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 18 2013

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@ #] == 1 &]; Table[d[n], {n, 1, 12}]
TableForm[%]
f[n_] := Length[Select[d[2 n], First[#] <= n &]]
Table[f[n], {n, 1, 20}]  (* A079122 *)
(* Clark Kimberling, Mar 13 2012 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-1]]]]; a[n_] := b[2*n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)
Table[SeriesCoefficient[Product[1 + x^(k/2), {k, 1, n}], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Jan 16 2024 *)

a(n) = b(0, n), b(m, n) = 1 + sum(b(i, j): m

Coefficient of x^(2*n) in Product_{k=1..n} (1+x^k). - Vladeta Jovovic, Aug 07 2003

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 22 2015

A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -3, -4, -3, -3, -1, -1, 2, 3, 7, 9, 14, 16, 23, 26, 33, 37, 45, 48, 57, 60, 68, 70, 77, 76, 82, 78, 80, 72, 70, 55, 48, 26, 11, -19, -42, -84, -116, -169, -213, -278, -333, -413, -479, -572, -651, -757, -846, -965, -1062

Offset: 0

Seiichi Manyama, Sep 08 2017

sign

			Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A035294, A278399, A278400, A278420, A292043, A292052.

N:= 100:
S := convert(series( add( (-1)^n*x^(n*(2*n+1))/(mul(1 - x^k,k = 1..2*n)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Re[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 08 2017 *)

( i*x; x)_inf is the g.f. for a(n) + i*A292043(n).
(-i*x; x)_inf is the g.f. for a(n) + i*A292052(n).
a(n)^2 + A292043(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017
From Peter Bala, Jan 15 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} (-1)^n*x^(n*(2*n+1))/Product_{k = 1..2*n}  (1 - x^k). Cf. A035294.
Conjectural g.f.: A(x) = (1/2)*Sum_{n >= 0} (-x)^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k). (End)

A079126 Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 2, 3, 4, 5, 0, 0, 0, 0, 1, 3, 4, 5, 6, 0, 0, 0, 0, 1, 3, 5, 6, 7, 8, 0, 0, 0, 0, 1, 3, 5, 7, 8, 9, 10, 0, 0, 0, 0, 0, 2, 5, 7, 9, 10, 11, 12, 0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15, 0, 0, 0, 0, 0, 1, 4, 8, 11, 13, 15, 16, 17, 18

Offset: 0

Reinhard Zumkeller, Dec 27 2002

T(n,n) = A000009(n), right side of the triangle;

T(n,k)=0 for n>0 and k < A002024(n); T(prime(n),n) = A067953(n) for n>0.

			The seven partitions of n=5 are {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1} and {1,1,1,1,1}. Only two of them ({4,1} and {3,2}) have distinct parts <= 4, so T(5,4) = 2.
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 0, 1, 2;
0, 0, 0, 1 ,2;
0, 0, 0, 1, 2, 3;
0, 0, 0, 1, 2, 3, 4;
0, 0, 0, 0, 2, 3, 4, 5;
0, 0, 0, 0, 1, 3, 4, 5,  6;
0, 0, 0, 0, 1, 3, 5, 6,  7,  8;
0, 0, 0, 0, 1, 3, 5, 7,  8,  9, 10;
0, 0, 0, 0, 0, 2, 5, 7,  9, 10, 11, 12;
0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15; ...

Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A000009, A079122, A035294, A079124, A079125.

Differs from A026840 in having extra zeros at the ends of the rows.

T:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, T(n, i-1)+`if`(i>n, 0, T(n-i, i-1))))
    end:
seq(seq(T(n,k), k=0..n), n=0..20);  # Alois P. Heinz, May 11 2015

T[n_, i_] := T[n, i] = If[n==0, 1, If[i<1, 0, T[n, i-1] + If[i>n, 0, T[n-i, i-1]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

T(n,k) = b(0,n,k), where b(m,n,k) = 1+sum(b(i,j,k): m

T(n,k) = coefficient of x^n in product_{i=1..k} (1+x^i). - Vladeta Jovovic, Aug 07 2003

A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 33, 45, 64, 87, 120, 159, 215, 283, 374, 486, 634, 814, 1049, 1335, 1700, 2146, 2708, 3390, 4243, 5276, 6552, 8095, 9989, 12266, 15044, 18375, 22409, 27235, 33049, 39974, 48281, 58148, 69923, 83871, 100452, 120027, 143214, 170515, 202731, 240567, 285073, 337195

Offset: 0

Robert G. Wilson v, Feb 24 2017

nonn

The even bisection of A282892. The other bisection is A078408.

			G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 10*x^8 + 16*x^9 + 22*x^10 + 33*x^11 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A035294, A035363, A078408, A214264, A282892.

with(numtheory):
b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
    end:
a:= n-> b(2*n, 0) -b(2*n, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[ f[2#] &, 52]
a[ n_] := SeriesCoefficient[ Sum[ Sign @ SquaresR[1, 16 k + 1] x^k, {k, n}] / QPochhammer[x], {x, 0, n}]; (* Michael Somos, Feb 24 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, n, issquare(16*k + 1)*x^k, A) / eta(x + A), n))}; /* Michael Somos, Feb 24 2017 */

a(n) = A282892(2n).
Expansion of (f(x^3, x^5) - 1) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 24 2017
a(n) = A035294(n) - A000041(n). - Michael Somos, Feb 24 2017

A343942 Number of even-length strict integer partitions of 2n+1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 52, 71, 96, 128, 170, 224, 292, 380, 491, 630, 805, 1024, 1295, 1632, 2049, 2560, 3189, 3959, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29249, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937, 172928

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

By conjugation, also the number of integer partitions of 2n+1 covering an initial interval of positive integers with greatest part even.

			The a(1) = 1 through a(7) = 13 strict partitions:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)      (7,6)      (8,7)
         (4,1)  (5,2)  (6,3)  (7,4)      (8,5)      (9,6)
                (6,1)  (7,2)  (8,3)      (9,4)      (10,5)
                       (8,1)  (9,2)      (10,3)     (11,4)
                              (10,1)     (11,2)     (12,3)
                              (5,3,2,1)  (12,1)     (13,2)
                                         (5,4,3,1)  (14,1)
                                         (6,4,2,1)  (6,4,3,2)
                                         (7,3,2,1)  (6,5,3,1)
                                                    (7,4,3,1)
                                                    (7,5,2,1)
                                                    (8,4,2,1)
                                                    (9,3,2,1)

Ranked by A005117 (strict), A028260 (even length), and A300063 (odd sum).
Odd bisection of A067661 (non-strict: A027187).
The non-strict version is A236914.
The opposite type of strict partition (odd length and even sum) is A344650.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A030229, A035294, A067659, A236559, A338907, A343941, A344649, A344654, A344739.

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,15}]

The Heinz numbers are A005117 /\ A028260 /\ A300063.

More terms from Bert Dobbelaere, Jun 12 2021

A385088 G.f.: Sum_{k>=0} x^k * Product_{j=1..2*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 13, 23, 39, 63, 101, 159, 243, 367, 547, 801, 1161, 1665, 2359, 3315, 4621, 6385, 8761, 11941, 16165, 21757, 29121, 38761, 51337, 67673, 88793, 116009, 150949, 195629, 252595, 324987, 416675, 532483, 678333, 861489, 1090913, 1377553, 1734761, 2178883

Offset: 0

Vaclav Kotesovec, Jun 17 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A035294, A192433.

Cf. A207641, A385089, A385090, A385091, A385092.

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 2*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(2*k))*(1 - x^(2*k - 1))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(2*k))*(1 + x^(2*k - 1))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n)) / (16 * n^(3/4)).

A249914 Number of partitions of 4n with equal sums of odd and even parts.

Original entry on oeis.org

1, 1, 4, 12, 30, 70, 165, 330, 704, 1380, 2688, 4984, 9394, 16665, 29970, 52096, 90090, 152064, 257180, 423360, 697851, 1129392, 1819632, 2891520, 4583250, 7162364, 11161752, 17211180, 26427544, 40208520, 60971520, 91641748, 137290956, 204198876, 302530560

Offset: 0

Alois P. Heinz, Feb 11 2015

nonn

			a(0) = 1: [], the empty partition.
a(1) = 1: [2,1,1].
a(2) = 4: [4,3,1], [4,1,1,1,1], [3,2,2,1], [2,2,1,1,1,1].
a(3) = 12: [6,5,1], [6,3,3], [6,3,1,1,1], [6,1,1,1,1,1,1], [5,4,2,1], [5,2,2,2,1], [4,3,3,2], [4,3,2,1,1,1], [4,2,1,1,1,1,1,1], [3,3,2,2,2], [3,2,2,2,1,1,1], [2,2,2,1,1,1,1,1,1].

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

Cf. A000009, A000041, A035294, A045931, A255001.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> combinat[numbpart](n) *b(2*n, 2*n-1):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]]];
a[n_] := PartitionsP[n] b[2n, 2n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = A000041(n) * A035294(n) = A000041(n) * A000009(2n).

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*6^(3/4)*n^(7/4)). - Vaclav Kotesovec, Dec 11 2020

A078406 Number of ways to partition 4*n into distinct positive integers.

Original entry on oeis.org

1, 2, 6, 15, 32, 64, 122, 222, 390, 668, 1113, 1816, 2910, 4582, 7108, 10880, 16444, 24576, 36352, 53250, 77312, 111322, 159046, 225585, 317788, 444793, 618784, 855906, 1177438, 1611388, 2194432, 2974400, 4013544, 5392550, 7215644

Offset: 0

N. J. A. Sloane, Dec 27 2002

Bisection of A035294. Cf. A078407.

a(n) = t(4*n, 0), t as defined in A079211.

Mathematica
```
Table[PartitionsQ[4n], {n, 0, 40}]
```

More terms from Don Reble, Jan 05 2003

A078410 Number of ways to partition 4*n+3 into distinct positive integers.

Original entry on oeis.org

2, 5, 12, 27, 54, 104, 192, 340, 585, 982, 1610, 2590, 4097, 6378, 9792, 14848, 22250, 32992, 48446, 70488, 101698, 145578, 206848, 291874, 409174, 570078, 789640, 1087744, 1490528, 2032290, 2757826, 3725410, 5010688, 6711480, 8953856

Offset: 0

N. J. A. Sloane, Dec 27 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Bisection of A078408. Cf. A035294, A000009, A078409.

PartitionsQ[4*Range[0,40]+3] (* Harvey P. Dale, Sep 23 2013 *)

a(n) = t(4*n+3, 0), t as defined in A079211.

More terms from Reinhard Zumkeller, Dec 28 2002

A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 3, 6, 11, 19, 33, 53, 86, 134, 205, 309, 460, 672, 974, 1394, 1975, 2773, 3863, 5333, 7316, 9964, 13484, 18140, 24269, 32288, 42751, 56331, 73888, 96503, 125529, 162635, 209939, 270027, 346123, 442213, 563205, 715110, 905361, 1142998, 1439098, 1807175

Offset: 0

Michael Somos, Oct 06 2015

nonn

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

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 33*x^6 + 53*x^7 + ...
G.f. = q^5 + q^21 + 3*q^37 + 6*q^53 + 11*q^69 + 19*q^85 + 33*q^101 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035294, A097451, A132217, A262160.

a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n}];
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x, -x^5]*f[x^3, x^5]/f[-x, -x^2]^2, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 31 2018 *)

{a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

Expansion of (psi(x^6) / psi(x) + psi(x^6) / psi(-x)) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [1, 2, 3, 2, 2, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 0, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 2, 3, 2, 1, 0, ...].
a(n) = A132217(2*n) = A262160(2*n).
Convolution product of A035294 and A097451.
a(n) ~ exp(sqrt(n)*Pi)/(8*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2015

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cp's OEIS Frontend

A079122 Number of ways to partition 2*n into distinct positive integers not greater than n.

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A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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A079126 Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.

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A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

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A343942 Number of even-length strict integer partitions of 2n+1.

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A385088 G.f.: Sum_{k>=0} x^k * Product_{j=1..2*k} (1 + x^j)/(1 - x^j).

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A249914 Number of partitions of 4n with equal sums of odd and even parts.

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