A001935 -id:A001935 - cp's OEIS frontend

A117148 Number of parts in all partitions of n in which no part occurs more than 3 times.

1, 3, 6, 8, 15, 24, 36, 50, 75, 102, 143, 197, 264, 349, 467, 606, 789, 1016, 1299, 1656, 2100, 2634, 3302, 4117, 5106, 6306, 7772, 9523, 11639, 14185, 17216, 20839, 25166, 30280, 36361, 43551, 52022, 62004, 73753, 87510, 103638, 122507, 144496, 170133

Offset: 1

Emeric Deutsch, Mar 07 2006

nonn

a(n) = sum(A117147(n,k), k>=1).

			a(4) = 8 because the partitions of 4 in which no part occurs more than 3 times are [4], [3,1], [2,2] and [2,1,1] ([1,1,1,1] does not qualify) with a total of 1+2+2+3=8 parts.

Vaclav Kotesovec, Table of n, a(n) for n = 1..15000 (terms 1..1000 from Alois P. Heinz)

Cf. A001935, A117147.

Column k=3 of A210485. - Alois P. Heinz, Jan 23 2013

g:=product(1+x^j+x^(2*j)+x^(3*j),j=1..55) *sum((x^j+2*x^(2*j)+3*x^(3*j))/ (1+x^j+x^(2*j)+x^(3*j)), j=1..55): gser:=series(g,x=0,53): seq(coeff(gser,x^n),n=1..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 3))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 08 2013

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[{l}, {l[[1]], l[[2]] + l[[1]]*j}][b[n-i*j, i-1]], {j, 0, Min[n/i, 3]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

G.f.: product(1+x^j+x^(2j)+x^(3j), j=1..infinity) * sum((x^j+2x^(2j)+3x^(3j)) / (1+x^j+x^(2j)+x^(3j)), j=1..infinity).

a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * n^(1/4)). - Vaclav Kotesovec, May 27 2018

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 30, 47, 76, 118, 181, 277, 417, 624, 929, 1367, 2001, 2913, 4210, 6056, 8665, 12328, 17466, 24640, 34600, 48395, 67442, 93625, 129520, 178588, 245429, 336252, 459324, 625613, 849762, 1151150, 1555378, 2096332, 2818630, 3780903, 5060240, 6757633, 9005106, 11975265

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A001935, A001936, A001937, A026007, A035528, A093160, A113415, A156616, A284474, A292037, A295832.

nmax = 44; CoefficientList[Series[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[x^k (1 - (-1)^k x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.
G.f.: exp(Sum_{k>=1} x^k*(1 - (-1)^k*x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3*(7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (12 * (7*Zeta(3))^(1/3)) - Pi^4 / (3024*Zeta(3)) - 1/24) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(11/8) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

Original entry on oeis.org

1, 1, 2, 4, 5, 8, 13, 17, 24, 36, 47, 64, 89, 115, 152, 204, 260, 336, 438, 552, 702, 896, 1117, 1400, 1758, 2171, 2688, 3332, 4079, 5000, 6131, 7446, 9048, 10992, 13255, 15984, 19264, 23081, 27644, 33084, 39408, 46912, 55797, 66107, 78264, 92572, 109140

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001935, A327046, A327047.

Cf. A107742, A327042.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 11^(1/4) * exp(sqrt(11*n/2)*Pi/3) / (2^(13/4)*sqrt(3)*n^(3/4)).

A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4*k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 21, 30, 45, 62, 85, 120, 161, 216, 293, 385, 505, 667, 862, 1112, 1438, 1833, 2330, 2965, 3733, 4688, 5887, 7334, 9114, 11319, 13970, 17203, 21162, 25905, 31643, 38605, 46911, 56891, 68904, 83179, 100224, 120603, 144719, 173360, 207396

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001935, A327045, A327047.

Cf. A107742, A327043.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(5) * exp(5*Pi*sqrt(n)/6) / (16*sqrt(3)*n^(3/4)).

A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5*k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 23, 34, 51, 72, 101, 143, 195, 267, 366, 487, 650, 866, 1135, 1487, 1940, 2504, 3226, 4145, 5283, 6714, 8513, 10725, 13481, 16905, 21085, 26244, 32588, 40299, 49732, 61229, 75131, 92004, 112435, 137009, 166627, 202269, 244919, 296038

Offset: 0

Vaclav Kotesovec, Aug 16 2019

nonn

In general, for fixed m>=1, if g.f. = Product_{k>=1} (Product_{j=1..m} (1 + x^(j*k))), then a(n) ~ HarmonicNumber(m)^(1/4) * exp(Pi*sqrt(HarmonicNumber(m)*n/3)) / (2^((m+3)/2) * 3^(1/4) * n^(3/4)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001935, A327045, A327046.

Cf. A107742, A327044.

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)) * (1+x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 137^(1/4) * exp(sqrt(137*n/5)*Pi/6) / (2^(9/2)*sqrt(3)*5^(1/4)*n^(3/4)).

A098491 Number of partitions of n with parts occurring at most thrice and an even number of parts. Row sums of A098489.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 6, 8, 11, 15, 19, 25, 32, 41, 52, 66, 83, 104, 129, 160, 197, 242, 296, 361, 438, 530, 640, 770, 923, 1105, 1318, 1569, 1864, 2208, 2611, 3082, 3628, 4264, 5003, 5858, 6848, 7993, 9312, 10833, 12584, 14595, 16904, 19552, 22582, 26049

Offset: 0

Ralf Stephan, Sep 12 2004

nonn

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Equals A001935 - A098492. Differs from A098492 at triangular indices.

a(n) = (A001935(n)+(-1)^n*A010054(n))/2. - Vladeta Jovovic, Sep 16 2004

More terms from Vladeta Jovovic, Sep 16 2004

A098492 Number of partitions of n with parts occurring at most thrice and an odd number of parts. Row sums of A098490.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 6, 8, 11, 14, 19, 25, 32, 41, 53, 66, 83, 104, 129, 160, 198, 242, 296, 361, 438, 530, 640, 769, 923, 1105, 1318, 1569, 1864, 2208, 2611, 3081, 3628, 4264, 5003, 5858, 6848, 7993, 9312, 10833, 12585, 14595, 16904, 19552, 22582, 26049

Offset: 0

Ralf Stephan, Sep 12 2004

nonn

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Equals A001935 - A098491. Differs from A098491 at triangular indices.

a(n) = (A001935(n)-(-1)^n*A010054(n))/2. - Vladeta Jovovic, Sep 16 2004

A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 13, 35, 87, 217, 539, 1291, 2999, 6880, 15595, 34738, 76202, 165282, 354655, 752546, 1580514, 3289337, 6787085, 13887937, 28195434, 56824772, 113729640, 226104615, 446665922, 877063515, 1712252521, 3324245063, 6419561961

Offset: 0

Paul D. Hanna, May 03 2010, May 08 2010

nonn

Compare to g.f. of partitions in which no parts are multiples of 4:

g.f. of A001935 = exp( Integral (theta_3(x)^4-1)/(8x) dx ).

			G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 35*x^4 + 87*x^5 +...
log(A(x)) = x + 7*x^2/2 + 28*x^3/3 + 71*x^4/4 + 126*x^5/5 +...+ A008457(n)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A138503, A008457, A001935, A000143, A307462.

nmax = 40; Abs[CoefficientList[Series[Product[1/(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Apr 10 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 - x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2019 *)

{a(n)=polcoeff(exp(sum(m=1,n, sumdiv(m,d,(-1)^(m-d)*d^3)*x^m/m)+x*O(x^n)),n)}

{a(n)=local(theta3=1+sum(m=1,sqrtint(2*n+2),2*x^(m^2)+x*O(x^n)));polcoeff(exp(intformal((theta3^8-1)/(16*x))),n)}

G.f.: exp( Sum_{n>=1} A008457(n)*x^n/n ) where A008457(n) = Sum_{d|n} (-1)^(n-d)*d^3.

a(n) ~ exp(2*Pi*n^(3/4)/3 - Zeta(3)/Pi^2) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 10 2019

A213598 Number of partitions of n in which no parts are multiples of 49.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173524

Offset: 0

Michael Somos, Jun 14 2012

nonn

For n<49 we have a(n)=A000041(n), for n>=49 a(n)!=A000041(n).

In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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.

Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A035959 (m=5), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10), A092885 (m=25), this sequence (m=49).

Cf. A000041, A287926.

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 13 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};

Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.
Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].
Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),
B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).
a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
a(n) = (1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017

A295342 The number of partitions of n in which at least one part is a multiple of 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 27, 37, 53, 71, 99, 131, 177, 232, 307, 397, 518, 663, 853, 1082, 1376, 1730, 2179, 2719, 3394, 4206, 5211, 6415, 7894, 9661, 11814, 14381, 17487, 21179, 25622, 30887, 37188, 44637, 53509, 63965, 76368, 90946, 108169, 128361

Offset: 0

R. J. Mathar, Nov 20 2017

nonn

Cf. A047967 (at least one multiple of 2), A295341 (at least one multiple of 3).

a(n) = A000041(n) - A001935(n).

cp's OEIS Frontend

A117148 Number of parts in all partitions of n in which no part occurs more than 3 times.

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A295831 Expansion of Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.

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A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)).

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A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)).

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A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) * (1 + x^(5*k)).

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A098491 Number of partitions of n with parts occurring at most thrice and an even number of parts. Row sums of A098489.

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A098492 Number of partitions of n with parts occurring at most thrice and an odd number of parts. Row sums of A098490.

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A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

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A213598 Number of partitions of n in which no parts are multiples of 49.

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A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3*k)).

A327046 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4*k)).

A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2k)) (1 + x^(3k)) (1 + x^(4k)) (1 + x^(5*k)).