A006906 -id:A006906 - cp's OEIS frontend

A265838 Expansion of Product_{k>=1} 1/(1 - k^4*x^k).

1, 1, 17, 98, 610, 2531, 18580, 72453, 449494, 2114440, 10753594, 48572844, 272867295, 1137441506, 5834448870, 27276382027, 129389072144, 576677550870, 2884567552542, 12401875640710, 59474089385344, 270438887909580, 1230979340265033, 5477371267093144

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1550

Cf. A006906, A077335, A265837, A265839, A265841.

Column k=4 of A292193.

nmax = 40; CoefficientList[Series[Product[1/(1 - k^4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(4*n/3), where
c = 27.2472595510480930563087281042486261391960582835336715327... if n mod 3 = 0
c = 26.8841208067599453033952496040472485838861626762931432887... if n mod 3 = 1
c = 26.9277867007233095885556073185206409643421012262073908850... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A265839 Expansion of Product_{k>=1} 1/(1 - k^5*x^k).

Original entry on oeis.org

1, 1, 33, 276, 2324, 13225, 145586, 760057, 6836328, 45996924, 322816122, 2064921330, 16881567137, 96217644312, 708147553326, 4769313137735, 31412238427954, 198869428043476, 1442034056253438, 8596120396405880, 58954590481229064, 387170921610808720

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1240

Cf. A006906, A077335, A265837, A265838, A265842.

Column k=5 of A292193.

nmax = 40; CoefficientList[Series[Product[1/(1 - k^5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(5*n/3), where
c = 12.8519823810391431573687005461910113782018563173082562291... if n mod 3 = 0
c = 12.4535903496941652158697054030067622653283880393322526099... if n mod 3 = 1
c = 12.5138855694494734654940524026530463555984202132997900068... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A285245 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 7, 10, 10, 10, 18, 24, 24, 24, 44, 56, 65, 65, 105, 129, 147, 147, 227, 292, 328, 355, 515, 645, 717, 771, 1107, 1367, 1562, 1670, 2429, 2949, 3339, 3555, 5073, 6181, 6961, 7546, 10582, 13059, 14619, 15789, 21925, 26886, 30235, 32575

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

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

Cf. A001156, A006906, A285243, A285244.

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

a(n) ~ c * 2^(n/4), where
c = 6.362854320457366874306510139107365081972383711876544726... if mod(n,4)=0
c = 6.470997903106304472752360748461108347899808941622559468... if mod(n,4)=1
c = 6.154059402265470959096395812318265046714869376472639022... if mod(n,4)=2
c = 5.624747659153211728892605407048217108787120474872434485... if mod(n,4)=3

A294609 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-jx^j)^(j^(kj)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 9, 6, 1, 1, 33, 90, 14, 1, 1, 129, 2220, 1154, 25, 1, 1, 513, 59178, 264908, 17427, 56, 1, 1, 2049, 1594836, 67176362, 49163017, 309117, 97, 1, 1, 8193, 43048770, 17181595604, 152662625259, 13120646934, 6285102, 198

Offset: 0

Seiichi Manyama, Nov 04 2017

			Square array begins:
    1,    1,      1,        1,           1, ...
    1,    1,      1,        1,           1, ...
    3,    9,     33,      129,         513, ...
    6,   90,   2220,    59178,     1594836, ...
   14, 1154, 264908, 67176362, 17181595604, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..2 give A006906, A294610, A294611.
Rows n=0-1 give A000012.
Cf. A294605.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k*d+1+j/d)) * A(n-j,k) for n > 0.

A306884 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) decreasing order.

Original entry on oeis.org

1, 1, 3, 6, 14, 28, 93, 270, 86170, 7625640881546

Offset: 0

Alois P. Heinz, Mar 15 2019

nonn

a(10) = 200352993...611306920 has 19729 decimal digits.

			a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
a(6) = 1^1^1^1^1^1 + 2^1^1^1^1 + 2^2^1^1 + 2^2^2 + 3^1^1^1 + 3^2^1 + 3^3 + 4^1^1 + 4^2 + 5^1 + 6 = 1 + 2 + 4 + 16 + 3 + 9 + 27 + 4 + 16 + 5 + 6 = 93.

Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Exponentiation
Wikipedia, Identity element
Wikipedia, Operator associativity
Wikipedia, Partition (number theory)

Cf. A006906, A066186, A306895, A306901, A306902, A306903, A306918.

f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(sort(l, `>`)), l=combinat[partition](n)):
seq(a(n), n=0..9);

A306895 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) increasing order.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 72, 387, 134349386, 115792089237316195423570985008687907853269984665640566457309223244801371506483

Offset: 0

Alois P. Heinz, Mar 15 2019

nonn

a(10) has 40403562 decimal digits.

			a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
a(6) = 1^1^1^1^1^1 + 1^1^1^1^2 + 1^1^2^2 + 2^2^2 + 1^1^1^3 + 1^2^3 + 3^3 + 1^1^4 + 2^4 + 1^5 + 6 = 1 + 1 + 1 + 16 + 1 + 1 + 27 + 1 + 16 + 1 + 6 = 72.

Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Exponentiation
Wikipedia, Identity element
Wikipedia, Operator associativity
Wikipedia, Partition (number theory)

Cf. A006906, A066186, A306884, A306901, A306902, A306903, A306919.

f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(sort(l, `<`)), l=combinat[partition](n)):
seq(a(n), n=0..9);

A318127 Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 2, 6, 19, 61, 191, 588, 1785, 5351, 15868, 46628, 135921, 393318, 1130538, 3229753, 9175347, 25931605, 72936434, 204223348, 569427145, 1581458917, 4375905243, 12065914843, 33160240020, 90848002909, 248154744196, 675932128695, 1836182233332, 4975249827916, 13447775233746

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

Binomial transform of A006906.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A006906, A218481, A294500.

a:=series(1/(1-x)*mul(1/(1-k*x^k/(1-x)^k),k=1..100),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019

nmax = 29; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[j^k x^(k j)/(k (1 - x)^(k j)), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 29}]

G.f.: (1/(1 - x))*exp(Sum_{k>=1} Sum_{j>=1} j^k*x^(k*j)/(k*(1 - x)^(k*j))).
a(n) = Sum_{k=0..n} binomial(n,k)*A006906(k).
a(n) ~ c * (1 + 3^(1/3))^n, where c = 97923.037496367052161042295948902147352859984491653037730624387144966464... = 1/((3^(1/3) - 1) * (3^(2/3) - 2)) * Product_{k>=4} 1/(1 - k/3^(k/3)). - Vaclav Kotesovec, Aug 19 2018

A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

Original entry on oeis.org

1, 3, 15, 54, 210, 699, 2484, 7995, 26610, 84186, 269940, 839238, 2634579, 8098194, 25032282, 76388265, 233791104, 709501596, 2157488730, 6523204836, 19747491810, 59558682132, 179762506329, 541222906812, 1630300772106, 4902697929306, 14748249476553

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

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

Cf. A006906, A265951, A265975, A265976.

Cf. A246935, A242587.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      3^n, b(n, i-1) +i*3*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

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

a(n) ~ c * 3^n, where c = Product_{m>=2} 1/(1 - m/3^(m-1)) = 5.86277744540963226378877460838259757442241952947887939654316926419876...

A265975 Expansion of Product_{k>=1} 1/(1 - 4kx^k).

Original entry on oeis.org

1, 4, 24, 108, 512, 2164, 9464, 39004, 163008, 663588, 2713752, 10954764, 44328512, 178160724, 716821752, 2874497660, 11532111232, 46187508676, 185028540696, 740595436652, 2964628293504, 11862432443764, 47467812675320, 189902835709212, 759756868215872

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

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

Cf. A006906, A265951, A265974, A265976.

Cf. A246935, A246936.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      4^n, b(n, i-1) +i*4*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-4*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = Product_{m>=2} 1/(1 - m/4^(m-1)) = 2.700170514502619666262858845683166558216386190684736249639219328278569...

A265976 Expansion of Product_{k>=1} 1/(1 - 5kx^k).

Original entry on oeis.org

1, 5, 35, 190, 1070, 5525, 29080, 147485, 752790, 3789170, 19105800, 95794930, 480650335, 2406018490, 12047084370, 60264282575, 301493182380, 1507758356660, 7540528037090, 37705593514220, 188545393000350, 942756783659980, 4713958620697385, 23570092258449540

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

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

Cf. A006906, A265951, A265974, A265975.

Cf. A246935, A246937.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      5^n, b(n, i-1) +i*5*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-5*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 5^n, where c = Product_{m>=2} 1/(1 - m/5^(m-1)) = 1.977268427518901757865749340705853730491796767544158844539130847296...

cp's OEIS Frontend

A265838 Expansion of Product_{k>=1} 1/(1 - k^4*x^k).

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A265839 Expansion of Product_{k>=1} 1/(1 - k^5*x^k).

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A285245 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2)).

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A294609 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j*x^j)^(j^(k*j)) in powers of x.

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A306884 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) decreasing order.

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A306895 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) increasing order.

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A318127 Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

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Maple

Mathematica

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A265974 Expansion of Product_{k>=1} 1/(1 - 3*k*x^k).

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Maple

Mathematica

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A265975 Expansion of Product_{k>=1} 1/(1 - 4*k*x^k).

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A294609 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-jx^j)^(j^(kj)) in powers of x.

A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

A265975 Expansion of Product_{k>=1} 1/(1 - 4kx^k).

A265976 Expansion of Product_{k>=1} 1/(1 - 5kx^k).