A238016 -id:A238016 - cp's OEIS frontend

A238610 Number of partitions of n^5 into parts that are at most n.

1, 1, 17, 5043, 7566280, 33432635477, 331281477244572, 6242342067484101895, 200386212932492140762672, 10133053906998476170548376403, 761287353202857218355451068558296, 81209207250425252838671771562105020103, 11863819168025774694177783984734986151951164

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Column k=5 of A238016.

a(n) = [x^(n^5)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(3*n-5) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238611 Number of partitions of n^6 into parts that are at most n.

Original entry on oeis.org

1, 1, 33, 44652, 478968264, 20735819929227, 2561606354507677872, 731267824140098782358035, 418829370245413954052212657987, 435014756168760380909523387186194290, 759593815557626617904440619008375351308296, 2102438285027143559802785522708457578986076133973

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Column k=6 of A238016.

a(n) = [x^(n^6)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(4*n-6) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238612 Number of partitions of n^7 into parts that are at most n.

Original entry on oeis.org

1, 1, 65, 399675, 30569959880, 12940003469288602, 19900384510848921094632, 85980297709044488588773397089, 877979540384895591800176962368065072, 18720322073127387624828552135278902045913865, 759424638305250205001161810310150848799911916053194

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Column k=7 of A238016.

a(n) = [x^(n^7)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(5*n-7) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238613 Number of partitions of n^8 into parts that are at most n.

Original entry on oeis.org

1, 1, 129, 3590508, 1955134763464, 8085018247233663602, 154721208025657067873668152, 10114611726199237476675435354424104, 1841159754991692001851990839259642586671980, 805821524592736878546553406787954567208757510893718

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Column k=8 of A238016.

a(n) = [x^(n^8)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(6*n-8) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238614 Number of partitions of n^9 into parts that are at most n.

Original entry on oeis.org

1, 1, 257, 32294883, 125107148059080, 5052825953808096554227, 1203080775953722005263023646232, 1189959092808570377265545326042454670975, 3861166489120966379893685013624485791901912419888, 34687845413783594101366282545316028561007822069601179170488

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Column k=9 of A238016.

a(n) = [x^(n^9)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(7*n-9) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238615 Number of partitions of n^10 into parts that are at most n.

Original entry on oeis.org

1, 1, 513, 290594892, 8006513870533064, 3157977415776418319210477, 9355115500676554620340590943203672, 139997247522791157386395916200494707946968395, 8097446373533819684208223226876398545717123633973546819

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

In general, for m > 3, is "Number of partitions of n^m into parts that are at most n" asymptotic to exp(2*n) * n^((m-2)*n-m) / (2*Pi). - Vaclav Kotesovec, May 25 2015

Alois P. Heinz, Table of n, a(n) for n = 0..60

Column k=10 of A238016.

a(n) = [x^(n^10)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(8*n-10) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238630 Number of partitions of 3^n into parts that are at most 3.

Original entry on oeis.org

1, 3, 12, 75, 588, 5043, 44652, 399675, 3590508, 32294883, 290594892, 2615176875, 23536060428, 211822949523, 1906401762732, 17157601515675, 154418370594348, 1389765206208963, 12507886468460172, 112570977053880075, 1013138789998136268, 9118249099522873203

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

			a(2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-39,27).

Row n=3 of A238016.

Cf. A001399, A051500.

gf:= (9*x^3+12*x^2-10*x+1)/((1-x)*(1-3*x)*(1-3^2*x)):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

Round[(3^Range[0, 25] + 3)^2/12] (* Paolo Xausa, Jun 26 2024 *)

a(n) = [x^(3^n)] Product_{j=1..3} 1/(1-x^j).
G.f.: (9*x^3+12*x^2-10*x+1)/((1-x)*(1-3*x)*(1-3^2*x)).
a(n) = A001399(3^n) = round((3^n+3)^2/12).
a(n) = 3*A051500(n-1) for n>=1. - Hugo Pfoertner, May 04 2024

A238631 Number of partitions of 4^n into parts that are at most 4.

Original entry on oeis.org

1, 5, 64, 2280, 123464, 7566280, 478968264, 30569959880, 1955134763464, 125107148059080, 8006513870533064, 512411390124519880, 32794241006913221064, 2098830017067059278280, 134325098574291643691464, 8596805948466686953550280, 550195574937260409780728264

Offset: 0

Alois P. Heinz, Mar 01 2014

Alois P. Heinz, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (85,-1428,5440,-4096).

Row n=4 of A238016.

gf:= -(2048*x^4+1460*x^3-1067*x^2+80*x-1)/((1-x)*(1-4*x)*(1-4^2*x)*(1-4^3*x)):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);

a(n) = [x^(4^n)] Product_{j=1..4} 1/(1-x^j).
G.f.: -(2048*x^4+1460*x^3-1067*x^2+80*x-1) / ((1-x) *(1-4*x) *(1-4^2*x) *(1-4^3*x)).
a(n) = (128 + 9*2^(3+2*n) + 15*16^n + 64^n)/144 for n > 0. - Stefano Spezia, Oct 08 2022

A238632 Number of partitions of 5^n into parts that are at most 5.

Original entry on oeis.org

1, 7, 377, 106852, 55567352, 33432635477, 20735819929227, 12940003469288602, 8085018247233663602, 5052825953808096554227, 3157977415776418319210477, 1973731034215692844676632352, 1233581290054852867292137569852, 770988230493054044846859764522977

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350

Row n=5 of A238016.

gf:= (1953125*x^5+4828125*x^4-2015125*x^3+96440*x^2-774*x+1)/mul(1-5^j*x, j=0..4):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);

a(n) = [x^(5^n)] Product_{j=1..5} 1/(1-x^j).

G.f.: (1953125*x^5+4828125*x^4-2015125*x^3+96440*x^2-774*x+1) / Product_{j=0..4} 1-5^j*x.

A238633 Number of partitions of 6^n into parts that are at most 6.

Original entry on oeis.org

1, 11, 2432, 6889527, 44056912182, 331281477244572, 2561606354507677872, 19900384510848921094632, 154721208025657067873668152, 1203080775953722005263023646232, 9355115500676554620340590943203672, 72745325498731282220397926627254957272

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Row n=6 of A238016.

gf:= -(29386561536*x^7 +220531481280*x^6 +188259164496*x^5 -77061923145*x^4 +2575778195*x^3 -12336681*x^2+9320*x-1)/ mul(1-6^j*x, j=0..5):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);

a(n) = [x^(6^n)] Product_{j=1..6} 1/(1-x^j).

G.f.: -(29386561536*x^7 +220531481280*x^6 +188259164496*x^5 -77061923145*x^4 +2575778195*x^3 -12336681*x^2 +9320*x-1) / Product_{j=0..5} 1-6^j*x.

cp's OEIS Frontend

A238610 Number of partitions of n^5 into parts that are at most n.

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A238611 Number of partitions of n^6 into parts that are at most n.

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A238612 Number of partitions of n^7 into parts that are at most n.

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A238613 Number of partitions of n^8 into parts that are at most n.

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A238614 Number of partitions of n^9 into parts that are at most n.

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A238615 Number of partitions of n^10 into parts that are at most n.

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A238630 Number of partitions of 3^n into parts that are at most 3.

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A238631 Number of partitions of 4^n into parts that are at most 4.

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A238632 Number of partitions of 5^n into parts that are at most 5.

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Maple

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A238633 Number of partitions of 6^n into parts that are at most 6.

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