A022629 -id:A022629 - cp's OEIS frontend

A318263 Expansion of Product_{k>=1} (1 + Lucas(k)*x^k).

1, 1, 3, 7, 11, 30, 62, 129, 235, 541, 1034, 2101, 4140, 8129, 15984, 31903, 60398, 117646, 228808, 433768, 836552, 1601282, 3031299, 5736396, 10899112, 20466182, 38556342, 72522116, 135662847, 253047629, 473785878, 878655661, 1634304062, 3033385668, 5608183925

Offset: 0

Vaclav Kotesovec, Aug 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Cf. A000032, A022629, A318248, A318264.

nmax = 40; CoefficientList[Series[Product[1+LucasL[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += LucasL[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

From Vaclav Kotesovec, Aug 24 2018: (Start)
a(n) ~ c * A000032(n) * A000009(n) ~ c * phi^n * exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)), where c = Product_{k>=1} ((1 + Lucas(k)/phi^k)/2) = 0.8503149035690839100210269103058319341315494385103929947491... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
Equivalently, c = Product_{k>=1} (1 + (-1)^k/(2*phi^(2*k))),
c = 2/3 * QPochhammer[-1/2, -1/GoldenRatio^2]. (End)

A325537 Irregular triangle whose rows are the sorted combined parts of all strict integer partitions of n.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 1, 2, 3, 4, 5, 1, 1, 2, 2, 3, 4, 5, 6, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10

Offset: 1

Gus Wiseman, May 08 2019

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with multiset union {1,1,2,2,3,4,5,6}, which is row n = 6.
Triangle begins:
  1
  2
  1 2 3
  1 3 4
  1 2 3 4 5
  1 1 2 2 3 4 5 6
  1 1 2 2 3 4 4 5 6 7
  1 1 1 2 2 3 3 4 5 5 6 7 8
  1 1 1 2 2 2 3 3 3 4 4 5 5 6 6 7 8 9

Row lengths are A015723.
Row sums are A066189.
Row products are A325504.
Run-lengths of row n are row n of A325513.
Cf. A000009, A006128, A022629, A181821, A302247, A325515.

Table[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,10}]

A326884 E.g.f.: Product_{k>=1} (1 + k*(exp(x)-1)^k).

Original entry on oeis.org

1, 1, 5, 43, 377, 4291, 58745, 914803, 15641897, 298104451, 6337624985, 147137420563, 3674045105417, 98093008751011, 2793940490888825, 84812168406518323, 2737609202984488937, 93486719521251467971, 3358396276982001106265, 126434158646122122080083

Offset: 0

Vaclav Kotesovec, Jul 31 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..410

Cf. A022629, A305550, A305987, A326885.

nmax = 20; CoefficientList[Series[Product[(1+k*(Exp[x]-1)^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} A022629(k)*Stirling2(n,k)*k!.

A022630 Expansion of Product_{m>=1} (1 + m*q^m)^2.

Original entry on oeis.org

1, 2, 5, 14, 28, 64, 133, 266, 513, 1000, 1873, 3420, 6257, 11078, 19585, 34192, 58714, 99870, 168858, 281666, 467082, 768994, 1253038, 2030658, 3269551, 5227868, 8304467, 13133256, 20630535, 32250274, 50181624, 77653530, 119634925, 183532470, 280245365

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=2 of A297321.

Cf. A022629, A022694.

Coefficients(&*[(1+m*x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018

nn=34; CoefficientList [Series[ Product[(1 + m*q^m)^2, {m, nn}], {q, 0, nn}],q] (* Robert G. Wilson v, Feb 08 2018 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^2)) \\ G. C. Greubel, Feb 16 2018

Self-convolution of A022629. - Alois P. Heinz, Dec 28 2017

G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A092485 Expansion of Product_{m>=1} (1+m(m+1)q^m).

Original entry on oeis.org

1, 2, 6, 24, 44, 142, 366, 800, 1636, 4338, 10154, 18968, 42368, 80726, 183914, 401096, 729944, 1402098, 2829814, 5172416, 10600836, 21582558, 37732782, 70148512, 127184636, 236798322, 416265730, 804045376, 1514022088, 2581172630, 4596614522, 8035151408

Offset: 0

Jon Perry, Apr 04 2004

nonn

Sum of product of i(i+1)-transform of terms in all partitions of n into distinct parts.

			The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding i(i+1)-transforms are of products 6*7, 2*5*6, 2*3*4*5, 2*2*3*3*4, so 42, 60, 120, 144 and their sum is a(6) = 366.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A022629, A265836.

Take[ CoefficientList[ Expand[ Product[1 + m(m + 1)q^m, {m, 1000}]], q], 30] (* Robert G. Wilson v, Apr 05 2004 *)

More terms from Robert G. Wilson v, Apr 05 2004

A268501 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k))^k.

Original entry on oeis.org

1, 0, 2, 6, 11, 32, 60, 148, 279, 690, 1312, 2778, 5684, 11282, 22920, 44724, 87919, 168978, 329800, 623086, 1189794, 2235744, 4189442, 7795642, 14438670, 26577246, 48616050, 88724110, 160629612, 290267100, 521225220, 933031364, 1661954928, 2950946220

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022629, A255528, A266891, A268500, A268502.

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

A268502 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k))^k.

Original entry on oeis.org

1, 2, 8, 26, 73, 210, 558, 1460, 3663, 9090, 21846, 51690, 120140, 274480, 618656, 1374792, 3017867, 6546610, 14053312, 29852658, 62825894, 131025056, 270948160, 555811298, 1131498850, 2286780266, 4589706604, 9151298134, 18131193484, 35706460678, 69910352496

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000219, A022629, A266891, A267004, A268500, A268501.

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

A269154 Expansion of Product_{k>=1} (1 + kx^k)/(1 + 2x^k).

Original entry on oeis.org

1, -1, 2, -3, 9, -13, 31, -53, 118, -210, 452, -866, 1793, -3493, 7119, -13992, 28257, -56253, 113035, -225318, 451745, -901870, 1805976, -3609701, 7222075, -14439594, 28887060, -57763494, 115540784, -231066845, 462154358, -924282660, 1848598423, -3697142099

Offset: 0

Vaclav Kotesovec, Feb 20 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..3300

Cf. A022629, A269144, A269153, A269155.

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

a(n) ~ c * (-2)^n, where c = Product_{k>=1} ((-2)^k + k)/((-2)^k - 1) = 0.4304067090888734207149852218007129877370867778815471457548443780472...

A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)^k).

Original entry on oeis.org

1, 1, 3, 12, 48, 211, 970, 4594, 22311, 110473, 555561, 2829918, 14570666, 75708835, 396481070, 2090558864, 11089276706, 59135014252, 316836936662, 1704764660218, 9207671377450, 49904141524184, 271325301723223, 1479427708380368, 8088057338101442, 44325245804200151

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 48*x^4 + 211*x^5 + 970*x^6 + 4594*x^7 + 22311*x^8 + 110473*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + 2*x^2*A(x)^2) * (1 + 3*x^3*A(x)^3) * (1 + 4*x^4*A(x)^4) * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 137*x^4/4 + 726*x^5/5 + 3896*x^6/6 + 21071*x^7/7 + 115089*x^8/8 + ... + A297322(n)*x^n/n + ...

Cf. A022629, A181315, A297322, A301456, A301577.

A306918 Sum over all partitions of n into distinct parts of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in decreasing order.

Original entry on oeis.org

1, 1, 2, 5, 7, 18, 36, 118, 265, 263212, 2217881, 152599933940, 542101086242752217003726400434973829461152534, 63340828764059520458379290673240751904836319648345

Offset: 0

Alois P. Heinz, Mar 16 2019

nonn

a(14) = 620606987...270037949 has 183231 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) = 3^2^1 + 4^2 + 5^1 + 6 = 9 + 16 + 5 + 6 = 36.

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

Cf. A022629, A066189, A306884, A306919.

d:= proc(l) local i; for i to nops(l)-1 do
       if l[i]=l[i+1] then return fi od; l
    end:
f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(l), l=map(l->d(sort(l, `>`)), combinat[partition](n))):
seq(a(n), n=0..13);

d[l_] := Module[{i}, For[i = 1, i <= Length[l] - 1, i++, If[l[[i]] == l[[i + 1]], Return[]]]; l];
f[l_] := If[l == {}, 1, l[[1]]^f[Delete[l, 1]]];
a[n_] := Sum[f[l], {l, ReverseSort /@ Select[IntegerPartitions[n], Length@# == Length@ Union@# &]}];
a /@ Range[0, 13] (* Jean-François Alcover, May 04 2020, after Maple *)

cp's OEIS Frontend

A318263 Expansion of Product_{k>=1} (1 + Lucas(k)*x^k).

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A325537 Irregular triangle whose rows are the sorted combined parts of all strict integer partitions of n.

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A326884 E.g.f.: Product_{k>=1} (1 + k*(exp(x)-1)^k).

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A022630 Expansion of Product_{m>=1} (1 + m*q^m)^2.

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A092485 Expansion of Product_{m>=1} (1+m*(m+1)*q^m).

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A268501 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k))^k.

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A268502 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k))^k.

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A269154 Expansion of Product_{k>=1} (1 + k*x^k)/(1 + 2*x^k).

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A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + k*x^k*A(x)^k).

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A092485 Expansion of Product_{m>=1} (1+m(m+1)q^m).

A269154 Expansion of Product_{k>=1} (1 + kx^k)/(1 + 2x^k).

A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)^k).