A300579 -id:A300579 - cp's OEIS frontend

A075900 Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).

1, 1, 3, 7, 19, 43, 115, 259, 659, 1523, 3731, 8531, 20883, 47379, 113043, 259219, 609683, 1385363, 3245459, 7344531, 17028499, 38579603, 88585619, 199845267, 457864595, 1028904339, 2339763603, 5256820115, 11896157587, 26626389395

Offset: 0

N. J. A. Sloane, Oct 15 2002

nonn

Number of compositions of partitions of n. a(3) = 7: 3, 21, 12, 111, 2|1, 11|1, 1|1|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split an integer composition of n into consecutive subsequences with weakly decreasing (or increasing) sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 19 splittings:
  ()  (1)  (2)      (3)          (4)
           (1,1)    (1,2)        (1,3)
           (1),(1)  (2,1)        (2,2)
                    (1,1,1)      (3,1)
                    (2),(1)      (1,1,2)
                    (1,1),(1)    (1,2,1)
                    (1),(1),(1)  (2,1,1)
                                 (2),(2)
                                 (3),(1)
                                 (1,1,1,1)
                                 (1,1),(2)
                                 (1,2),(1)
                                 (2),(1,1)
                                 (2,1),(1)
                                 (1,1),(1,1)
                                 (1,1,1),(1)
                                 (2),(1),(1)
                                 (1,1),(1),(1)
                                 (1),(1),(1),(1)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..3180 (terms 0..1000 from Alois P. Heinz)
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Row sums of A327549.
The strict case is A304961.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Splittings with distinct sums are A336127.
Starting with a reversed partition gives A316245.
Starting with a partition instead of composition gives A336136.
Cf. A006951, A063834, A300579, A317715, A319794, A323582, A327548, A336135.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(&*[1-2^(j-1)*x^j: j in [1..m+2]]) )); // G. C. Greubel, Jan 25 2024

oo := 101; t1 := mul(1/(1-x^n/2),n=1..oo): t2 := series(t1,x,oo-1): t3 := seriestolist(t2): A075900 := n->2^n*t3[n+1];
with(combinat); A075900 := proc(n) local i,t1,t2,t3; t1 := partition(n); t2 := 0; for i from 1 to nops(t1) do t3 := t1[i]; t2 := t2+2^(n-nops(t3)); od: t2; end;

b[n_]:= b[n]= Sum[d*2^(n - n/d), {d, Divisors[n]}];
a[0]= 1; a[n_]:= a[n]= 1/n*Sum[b[k]*a[n-k], {k,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Mar 20 2014, after Vladeta Jovovic, fixed by Vaclav Kotesovec, Mar 08 2018 *)

s(m,n):=if nVladimir Kruchinin, Sep 06 2014 */

{a(n)=polcoeff(prod(k=1,n,1/(1-2^(k-1)*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Jan 13 2013

{a(n)=polcoeff(exp(sum(k=1,n+1,x^k/(k*(1-2^k*x^k)+x*O(x^n)))),n)} \\ Paul D. Hanna, Jan 13 2013

m=80;
def A075900_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/product(1-2^(j-1)*x^j for j in range(1,m+1)) ).list()
A075900_list(m) # G. C. Greubel, Jan 25 2024

a(n) = Sum_{ partitions n = c_1 + ... + c_k } 2^(n-k). If p(n, m) = number of partitions of n into m parts, a(n) = Sum_{m=1..n} p(n, m)*2^(n-m).
G.f.: Sum_{n>=0} (a(n)/2^n)*x^n = Product_{n>0} 1/(1-x^n/2). - Vladeta Jovovic, Feb 11 2003
a(n) = 1/n*Sum_{k=1..n} A080267(k)*a(n-k). - Vladeta Jovovic, Feb 11 2003
G.f.: exp( Sum_{n>=1} x^n / (n*(1 - 2^n*x^n)) ). - Paul D. Hanna, Jan 13 2013
a(n) = s(1,n), a(0)=1, where s(m,n) = Sum_{k=m..n/2} 2^(k-1)*s(k, n-k)  + 2^(n-1), s(n,n) = 2^(n-1), s(m,n)=0, m>. - Vladimir Kruchinin, Sep 06 2014
a(n) ~ 2^(n-2) * (Pi^2 - 6*log(2)^2)^(1/4) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (3^(1/4) * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Mar 09 2018

More terms from Vladeta Jovovic, Feb 11 2003

A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 12, 36, 135, 432, 1539, 4860, 17496, 55404, 192456, 623295, 2125764, 6849684, 23442453, 75110328, 252965916, 822670668, 2735858268, 8838926712, 29501352792, 95090206689, 314068876416, 1018141045092, 3342663979092, 10798571289897, 35481518064576

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A032308, A300579, A304961, A340103, A344063, A344064, A344065, A344066, A344067, A344068.

nmax = 27; CoefficientList[Series[Product[(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 27}]

seq(n)={Vec(prod(k=1, n, 1 + 3^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/3))^(1/4) * 3^n * exp(2*sqrt(-polylog(2, -1/3)*n)) / (4*sqrt(Pi/3)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338673 Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).

Original entry on oeis.org

1, 1, 5, 21, 101, 421, 2021, 8421, 39397, 167397, 766437, 3244517, 14881253, 62804453, 283415013, 1210159589, 5401907685, 22966866405, 102497423845, 435085808101, 1925197238757, 8215432696293, 36068400468453, 153579729097189, 674546796630501, 2866238341681637, 12508012102193637

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246936, A300579, A338674, A338675, A338676, A338677, A338678, A338679.

nmax = 26; CoefficientList[Series[Product[1/(1 - 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 4^(n - k), {k, 0, n}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

a(n) = Sum_{k=0..n} p(n,k) * 4^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(3) * polylog(2, 1/4)^(1/4) * 4^(n - 1/2) * exp(2*sqrt(polylog(2, 1/4)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338674 Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).

Original entry on oeis.org

1, 1, 6, 31, 181, 931, 5431, 27931, 159806, 834806, 4697306, 24478556, 137931681, 717306681, 3989650431, 20958791056, 115494337931, 604881056681, 3333662306681, 17439531447306, 95396181837931, 501716543166056, 2725636758009806, 14311071572462931, 77793648720900431

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246937, A300579, A338673, A338675, A338676, A338677, A338678, A338679.

nmax = 24; CoefficientList[Series[Product[1/(1 - 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 5^(n - k), {k, 0, n}], {n, 0, 24}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

a(n) = Sum_{k=0..n} p(n,k) * 5^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ polylog(2, 1/5)^(1/4) * 5^(n - 1/2) * exp(2*sqrt(polylog(2, 1/5)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338675 Expansion of Product_{k>=1} 1 / (1 - 6^(k-1)*x^k).

Original entry on oeis.org

1, 1, 7, 43, 295, 1807, 12391, 75895, 512647, 3179815, 21196807, 131258311, 875934727, 5416216711, 35763798535, 223059458311, 1461247179271, 9093600322567, 59586011601415, 370499158291975, 2411884242270727, 15072418547458567, 97530161503173127, 608700350537722375

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246938, A300579, A338673, A338674, A338676, A338677, A338678, A338679.

nmax = 23; CoefficientList[Series[Product[1/(1 - 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 6^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) = Sum_{k=0..n} p(n,k) * 6^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(5) * polylog(2, 1/6)^(1/4) * 6^(n - 1/2) * exp(2*sqrt(polylog(2, 1/6)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338676 Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).

Original entry on oeis.org

1, 1, 8, 57, 449, 3193, 25145, 178809, 1391314, 9996498, 76955586, 552257546, 4255024523, 30502987019, 232969386483, 1682476714724, 12762937304013, 92019035596293, 698222541789109, 5030814634614406, 37955614705675479, 274741644961416648, 2061916926761604144, 14909943849253537057

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246939, A300579, A338673, A338674, A338675, A338677, A338678, A338679.

nmax = 23; CoefficientList[Series[Product[1/(1 - 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 7^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) = Sum_{k=0..n} p(n,k) * 7^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(6) * polylog(2, 1/7)^(1/4) * 7^(n - 1/2) * exp(2*sqrt(polylog(2, 1/7)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338677 Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).

Original entry on oeis.org

1, 1, 9, 73, 649, 5257, 46729, 378505, 3331721, 27219593, 237491849, 1938544265, 16925054601, 138041874057, 1196384310921, 9820024329865, 84609648809609, 693596417152649, 5977550934234761, 48976660041553545, 419984680697190025, 3455551232025810569, 29494747047731910281

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246940, A300579, A338673, A338674, A338675, A338676, A338678, A338679.

nmax = 22; CoefficientList[Series[Product[1/(1 - 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 8^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) = Sum_{k=0..n} p(n,k) * 8^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(7) * polylog(2, 1/8)^(1/4) * 8^(n - 1/2) * exp(2*sqrt(polylog(2, 1/8)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338678 Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).

Original entry on oeis.org

1, 1, 10, 91, 901, 8191, 81091, 737191, 7239142, 66288142, 646149322, 5912729632, 57664985653, 527352541453, 5111015223223, 46998961540624, 453182267869615, 4163124744738505, 40151590267580785, 368699990679135946, 3540322181970716707, 32632895079429817528, 312061810101214595698

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

Cf. A008284, A075900, A246941, A300579, A338673, A338674, A338675, A338676, A338677, A338679.

nmax = 22; CoefficientList[Series[Product[1/(1 - 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 9^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) = Sum_{k=0..n} p(n,k) * 9^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(2) * polylog(2, 1/9)^(1/4) * 9^(n - 1/2) * exp(2*sqrt(polylog(2, 1/9)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338679 Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k).

Original entry on oeis.org

1, 1, 11, 111, 1211, 12211, 133211, 1343211, 14553211, 147653211, 1589753211, 16120753211, 173641753211, 1759951753211, 18855161753211, 192028261753211, 2048080361753211, 20841811361753211, 222333332361753211, 2261780642361753211, 24033895852361753211, 245331468952361753211

Offset: 0

Ilya Gutkovskiy, Apr 23 2021

nonn

In general, if g.f. = Product_{k>=1} 1/(1 - d^(k-1)*x^k), where d > 1, then a(n) ~ sqrt(d-1) * polylog(2, 1/d)^(1/4) * d^(n - 1/2) * exp(2*sqrt(polylog(2, 1/d)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Cf. A008284, A075900, A246942, A300579, A338673, A338674, A338675, A338676, A338677, A338678.

nmax = 21; CoefficientList[Series[Product[1/(1 - 10^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 10^(n - k), {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) = Sum_{k=0..n} p(n,k) * 10^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ 3 * polylog(2, 1/10)^(1/4) *10^(n - 1/2) * exp(2*sqrt(polylog(2, 1/10)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338697 a(n) = [x^n] Product_{k>=1} 1 / (1 - n^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 13, 101, 931, 12391, 178809, 3331721, 66288142, 1589753211, 40104031166, 1183380156013, 36187564837217, 1262524447510383, 45533370885563716, 1834219414937219601, 76016894083755947753, 3479900167920331954531, 162982921698852088968886, 8341707623665223127224821

Offset: 0

Ilya Gutkovskiy, Apr 24 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Cf. A008284, A075900, A124577, A300579, A338673, A338674, A338675, A338676, A338677, A338678, A338679, A344095.

Table[SeriesCoefficient[Product[1/(1 - n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Length[IntegerPartitions[n, {k}]] n^(n - k), {k, 0, n}], {n, 1, 20}]]
Join[{1}, Table[SeriesCoefficient[x + (n-1)/(n*QPochhammer[1/n, n*x]), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

a(n) = Sum_{k=0..n} p(n,k) * n^(n-k), where p(n,k) is the number of partitions of n into k parts.

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021

cp's OEIS Frontend

A075900 Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).

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A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

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A338673 Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).

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A338674 Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).

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A338675 Expansion of Product_{k>=1} 1 / (1 - 6^(k-1)*x^k).

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A338676 Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).

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A338677 Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).

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A338678 Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).

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A338679 Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k).

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