A264686 -id:A264686 - cp's OEIS frontend

A268498 Expansion of Product_{k>=1} ((1 + 2*x^k) / (1 + x^k)).

1, 1, 0, 3, -1, 3, 3, 3, 0, 4, 12, 0, 9, -3, 21, 12, 17, -3, 33, 0, 33, 36, 36, 27, 21, 52, 24, 90, 72, 99, 24, 138, 21, 207, 0, 261, 149, 267, 45, 333, 174, 339, 174, 345, 411, 654, 330, 456, 657, 535, 684, 483, 1233, 489, 1353, 882, 1803, 720, 1902, 756

Offset: 0

Vaclav Kotesovec, Feb 06 2016

sign

It appears that this sequence contains only finitely many nonpositive terms, namely at indices {2, 4, 8, 11, 13, 17, 19, 34}. - Gus Wiseman, Jan 23 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A032302, A264686, A268499, A268500.

Cf. A006951, A100471, A133121.

nmax = 100; CoefficientList[Series[Product[(1+2*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(2)^2 + 4*polylog(2, -1/2) = 2.4571173338382709125... .
a(n) = Sum_{k = 0...n} (-1)^k * A133121(n,k). - Gus Wiseman, Jan 23 2019
G.f.: Product_{k>=1} (1 - Sum_{j>=1} (-1)^j * x^(k*j)). - Ilya Gutkovskiy, Nov 06 2019

A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 8, 5, 0, 1, 5, 8, 15, 14, 7, 0, 1, 6, 10, 24, 27, 24, 11, 0, 1, 7, 12, 35, 44, 51, 40, 15, 0, 1, 8, 14, 48, 65, 88, 93, 64, 22, 0, 1, 9, 16, 63, 90, 135, 176, 159, 100, 30, 0, 1, 10, 18, 80, 119, 192, 295, 312, 264, 154, 42, 0

Offset: 0

Alois P. Heinz, Aug 27 2019

			A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,    1,    1,    1, ...
  0,  1,   2,   3,   4,   5,    6,    7,    8, ...
  0,  2,   4,   6,   8,  10,   12,   14,   16, ...
  0,  3,   8,  15,  24,  35,   48,   63,   80, ...
  0,  5,  14,  27,  44,  65,   90,  119,  152, ...
  0,  7,  24,  51,  88, 135,  192,  259,  336, ...
  0, 11,  40,  93, 176, 295,  456,  665,  928, ...
  0, 15,  64, 159, 312, 535,  840, 1239, 1744, ...
  0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Jessica Jay and Benjamin Lees, Combinatorial identities from an inhomogeneous Ising chain, arXiv:2401.16311 [math.PR], 2024.
Wikipedia, Partition (number theory)

Columns k=0-4 give: A000007, A000041, A015128, A264686, A266821.
Main diagonal gives A321880.
Cf, A000142, A003056, A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} (1+(k-1)*x^j)/(1-x^j).

A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).

A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

Original entry on oeis.org

1, 3, 9, 24, 60, 141, 324, 717, 1560, 3330, 7020, 14622, 30225, 61998, 126522, 257007, 520326, 1050396, 2116116, 4255584, 8547330, 17149350, 34382295, 68889840, 137969466, 276220962, 552865365, 1106356314, 2213644548, 4428657402, 8859340926, 17721640698

Offset: 0

Vaclav Kotesovec, Nov 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A006951, A070933, A261584, A264686.

nmax = 40; CoefficientList[Series[Product[(1 + x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 2^n, where c = A079555 / A048651 = Product_{k>=1} (2^k+1)/(2^k-1) = 8.25598793577825006554414084943227312652...

A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

Original entry on oeis.org

1, 4, 8, 24, 44, 88, 176, 312, 544, 924, 1584, 2552, 4136, 6488, 10128, 15632, 23748, 35640, 53080, 78136, 114024, 165552, 237744, 339544, 481248, 678236, 949008, 1321840, 1830376, 2521688, 3456672, 4717208, 6406680, 8666448, 11672464, 15660528, 20934868

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A000041 and A032308.

In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 - x^k)) then a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(m+1)*n), where c = 2*Pi^2/3 + log(m)^2 + 2*polylog(2, -1/m).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5001 terms from Vaclav Kotesovec)

Cf. A000041, A015128, A032308, A264686.

Column k=4 of A321884.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1)))(n-i*j), j=1..n/i)*4 +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..44);  # Alois P. Heinz, Aug 28 2019

nmax = 40; CoefficientList[Series[Product[(1+3*x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

{ my(n=40); Vec(prod(k=1, n, 4/(1-x^k) - 3 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (8*Pi*n), where c = 2*Pi^2/3 + log(3)^2 + 2*polylog(2, -1/3) = 7.16861897522987077909937377164783326088308015803... .

A347830 a(n) = Sum_{k=0..n} 2^k * A000009(k) * A000041(n-k).

Original entry on oeis.org

1, 3, 8, 27, 67, 189, 509, 1329, 3344, 8694, 22062, 54756, 136741, 335103, 822277, 2016738, 4872787, 11711655, 28253743, 67319328, 160333627, 381350646, 901272326, 2121969771, 4991176893, 11689645776, 27305992220, 63705989106, 148106539514, 343371565449, 795524336390

Offset: 0

Vaclav Kotesovec, Sep 15 2021

nonn

Cf. A000009, A000041, A015128, A065446, A264686, A347829.

Table[Sum[2^k*PartitionsQ[k]*PartitionsP[n-k], {k, 0, n}], {n, 0, 50}]
nmax = 50; CoefficientList[Series[Product[(1 + 2^k*x^k) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ A065446 * 2^n * A000009(n).
a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/3)) / (3^(1/4) * QPochhammer(1/2) * n^(3/4)).
G.f.: Product_{k>=1} (1 + 2^k*x^k) / (1 - x^k).

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A268498 Expansion of Product_{k>=1} ((1 + 2*x^k) / (1 + x^k)).

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A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

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A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

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A347830 a(n) = Sum_{k=0..n} 2^k * A000009(k) * A000041(n-k).

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