A022567 -id:A022567 - cp's OEIS frontend

A329289 Expansion of g.f. (1 + x) * (1 + x^2) * Product_{k>=1} (1 + x^k).

1, 2, 3, 5, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 82, 99, 119, 143, 170, 202, 240, 283, 333, 391, 457, 533, 621, 721, 835, 966, 1114, 1282, 1474, 1690, 1935, 2213, 2525, 2877, 3274, 3719, 4219, 4781, 5409, 6112, 6900, 7778, 8758, 9852, 11068, 12422

Offset: 0

Ilya Gutkovskiy, Jun 07 2020

Number of partitions of n into distinct parts if there are two types of 1's and two types of 2's.

Cf. A000009, A000097, A022567, A036469, A052816, A096914, A329384.

series(1/2 * add( x^((n-2)*(n-3)/2) / mul(1 - x^k, k = 1..n), n = 0..12), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Feb 03 2025

nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 3, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]

a(n) = A036469(n) - A036469(n-4).
a(n) ~ exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 11 2020
G.f.: A(x) = 1/2 * Sum_{n >= 0} x^((n-2)*(n-3)/2) / (Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025

A192096 Maximum number of tatami tilings of any m X m square region with exactly n horizontal dimers and m monomers.

Original entry on oeis.org

2, 4, 6, 12, 18, 28, 44, 64, 92, 132, 186, 256, 352, 476, 638, 852, 1124, 1472, 1920, 2484, 3196, 4096, 5216, 6612, 8350, 10496, 13140, 16396, 20380, 25244, 31178, 38380, 47104, 57660, 70380, 85684, 104068, 126080, 152396, 183808, 221208, 265664, 318432

Offset: 0

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 15 2011

nonn

A tatami tiling consists of dimers (1 X 2) and monomers (1 X 1) where no four meet at a point.

			a(0) = 2 because exactly 2 tilings are possible for 0 horizontal dimers and any m >= 2.  For example, with m = 3:
    _ _ _      _ _ _
   |_| |_|    | |_| |
   | |_| |    |_| |_|
   |_|_|_|    |_|_|_|

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Monomer-Dimer Tatami Tilings of Rectangular Regions, Electronic Journal of Combinatorics, 18(1) (2011) P109, 24 pages.

Cf. A192095, A022567.

gf:= n-> 2 * mul((1 + x^k)^2, k=1..n):
a:= n-> coeff(series(gf(n), x, n+1), x, n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 15 2011

G.f.: 2 * Product_{k>0} (1 + x^k)^2.

a(n) = 2 * A022567(n).

A300417 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^2.

Original entry on oeis.org

1, 2, 1, 2, 4, 2, 3, 6, 3, 4, 10, 8, 5, 10, 11, 8, 14, 16, 11, 18, 22, 18, 23, 22, 22, 34, 31, 26, 39, 40, 33, 50, 56, 36, 53, 74, 51, 62, 86, 68, 77, 98, 86, 88, 102, 106, 120, 130, 120, 136, 157, 134, 157, 194, 155, 182, 241, 194, 196, 256, 237, 236, 288, 282, 273, 324

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Number of partitions of n into distinct triangular parts (A000217), with 2 types of each part.

Self-convolution of A024940.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000217, A022567, A024940, A279226, A298435.

nmax = 65; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000217(k))^2.

a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2)/2)^(2/3) * n^(1/3)) * ((sqrt(2)-1) * Zeta(3/2) / (2*Pi))^(1/3) / (4*sqrt(3) * n^(5/6)). - Vaclav Kotesovec, Mar 05 2018

A329384 G.f.: (1 + x) * (1 + x^2) * (1 + x^3) * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 16, 20, 26, 34, 43, 54, 68, 84, 103, 127, 154, 186, 225, 269, 321, 383, 453, 535, 631, 740, 866, 1012, 1178, 1368, 1587, 1835, 2117, 2440, 2804, 3217, 3687, 4215, 4812, 5487, 6244, 7096, 8055, 9128, 10331, 11681, 13187, 14870, 16752, 18846, 21180

Offset: 0

Ilya Gutkovskiy, Jun 07 2020

nonn

Number of partitions of n into distinct parts if there are two types of 1's, two types of 2's and two types of 3's.

Cf. A000009, A000098, A022567, A036469, A052816, A329289.

nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) (1 + x^3) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 4, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]

a(n) = A036469(n) + A036469(n-3) - A036469(n-4) - A036469(n-7).

a(n) ~ 2*exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 11 2020

A343204 Numerators of coefficients in expansion of Product_{k>=1} (1 + x^k)^(1/2).

Original entry on oeis.org

1, 1, 3, 13, 67, 239, 1031, 2501, 36579, 109915, 468653, 1043851, 9395751, 21232827, 97493519, 235880373, 7717800611, 17385733651, 82456426833, 175398844079, 1578297716013, 3634938193489, 15867173716609, 34517119775523, 619312307079687, 1363237700933583

Offset: 0

Ilya Gutkovskiy, Apr 07 2021

			1, 1/2, 3/8, 13/16, 67/128, 239/256, 1031/1024, 2501/2048, 36579/32768, 109915/65536, 468653/262144, 1043851/524288, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1631

Cf. A000009, A022567, A046161 (denominators), A061159, A098987.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d/2, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021

nmax = 25; CoefficientList[Series[Product[(1 + x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator
a[n_] := a[n] = If[n == 0, 1, (1/(2 n)) Sum[Sum[Mod[d, 2] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}] // Numerator

a(n) / A046161(n) ~ exp(sqrt(n/6)*Pi) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Apr 12 2021

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 10, 39, 390, 1521, 7830, 49518, 207360, 951102, 4264650, 22185657, 89579520, 401428224, 1676401110, 7172977275, 31972081050, 130330236546, 537393139200, 2213787635712, 8988968449530, 36073295687070, 150459195064320, 590262148332288, 2362876271009370, 9314694641056095

Offset: 0

Vaclav Kotesovec, Mar 02 2024

nonn

In general, if d >= 1 and g.f. = Product_{k>=1} (1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k), then a(n) ~ d^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(d)^2))) * (Pi^2/3 + log(d)^2)^(1/4) / (2^(5/4) * sqrt(Pi) * (d+1) * n^(3/4)).

Cf. A032308, A344062.

Cf. A022567 (d=1), A370761 (d=2).

nmax = 25; CoefficientList[Series[Product[(1+3^(k+1)*x^k)*(1+3^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(3)^2))) * (Pi^2/3 + log(3)^2)^(1/4) / (2^(13/4) * sqrt(Pi) * n^(3/4)).

A374018 Expansion of Product_{k>=1} 1 / (1 - x^(3*k-1))^2.

Original entry on oeis.org

1, 0, 2, 0, 3, 2, 4, 4, 7, 6, 13, 10, 19, 18, 27, 30, 42, 44, 63, 66, 91, 100, 130, 144, 187, 206, 263, 294, 364, 412, 506, 568, 696, 782, 943, 1070, 1273, 1444, 1713, 1936, 2285, 2586, 3027, 3428, 3996, 4516, 5243, 5924, 6841, 7730, 8895, 10030, 11512, 12966, 14825, 16696

Offset: 0

Ilya Gutkovskiy, Jun 25 2024

nonn

Cf. A000712, A022567, A035386, A078182, A261616, A374019.

nmax = 55; CoefficientList[Series[Product[1/(1 - x^(3 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A078182(k) * a(n-k).
a(n) = Sum_{k=0..n} A035386(k) * A035386(n-k).
a(n) ~ exp(2*Pi*sqrt(n)/3) * Pi^(4/3) / (3^(3/2) * Gamma(1/3)^2 * n^(11/12)). - Vaclav Kotesovec, Jun 25 2024

A374019 Expansion of Product_{k>=1} 1 / (1 - x^(4*k-1))^2.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 3, 2, 0, 4, 4, 2, 5, 6, 7, 8, 8, 12, 15, 12, 17, 26, 23, 24, 37, 40, 39, 50, 62, 66, 74, 86, 101, 116, 122, 144, 175, 184, 202, 246, 274, 294, 340, 388, 432, 480, 533, 610, 684, 742, 835, 956, 1045, 1144, 1299, 1450, 1586, 1758, 1965, 2182, 2400, 2638, 2941, 3268, 3560, 3922

Offset: 0

Ilya Gutkovskiy, Jun 25 2024

nonn

Cf. A000712, A022567, A035462, A050452, A261629, A374018.

nmax = 65; CoefficientList[Series[Product[1/(1 - x^(4 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A050452(k) * a(n-k).
a(n) = Sum_{k=0..n} A035462(k) * A035462(n-k).
a(n) ~ Pi^(3/2) * exp(Pi*sqrt(n/3)) / (2*sqrt(3) * Gamma(1/4)^2 * n). - Vaclav Kotesovec, Jun 25 2024

cp's OEIS Frontend

A329289 Expansion of g.f. (1 + x) * (1 + x^2) * Product_{k>=1} (1 + x^k).

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A192096 Maximum number of tatami tilings of any m X m square region with exactly n horizontal dimers and m monomers.

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A300417 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^2.

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A329384 G.f.: (1 + x) * (1 + x^2) * (1 + x^3) * Product_{k>=1} (1 + x^k).

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A343204 Numerators of coefficients in expansion of Product_{k>=1} (1 + x^k)^(1/2).

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A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)*x^k) * (1 + 3^(k-1)*x^k).

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A374018 Expansion of Product_{k>=1} 1 / (1 - x^(3*k-1))^2.

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A374019 Expansion of Product_{k>=1} 1 / (1 - x^(4*k-1))^2.

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A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).