A162663 -id:A162663 - cp's OEIS frontend

A036081 The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.

1, 2, 16, 202, 3044, 52794, 1055260, 24081754, 615896308, 17347970202, 531721375308, 17595339114554, 624882463734756, 23691503493287738, 954301756159098172, 40665568780962213530, 1826521141853468785364

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074.

Column 11 of A162663.

u[0, j_] := 1; u[k_, j_] := u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 11], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 - after Wouter Meeussen in similar sequences *)
mx = 16; p = 11; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 11^k * BellB[k, 1/11] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=11.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=11. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (11*n/LambertW(11*n))^n * exp(n/LambertW(11*n) + (11*n/LambertW(11*n))^(1/11) - n - 12/11) / sqrt(1 + LambertW(11*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A141005 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=8.

Original entry on oeis.org

1, 4, 31, 329, 4316, 66543, 1172077, 23139068, 504673027, 12023659317, 310112290044, 8596075462411, 254551364575929, 8012189624924740, 266901629110149847, 9374566215360038977, 346044223246303101068

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=8 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 8], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 *)

A141006 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=9.

Original entry on oeis.org

1, 3, 22, 235, 3139, 49376, 891491, 18160071, 411124024, 10212362573, 275543711899, 8012352809874, 249516568317597, 8277748828228015, 291219448755079278, 10821916875914147551, 423314732431005599479

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=9 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 9], {n, 0, 30}](* Vincenzo Librandi, Dec 12 2012 *)

A141007 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=10.

Original entry on oeis.org

1, 4, 34, 410, 6170, 109506, 2227882, 50987378, 1293386410, 35933906946, 1083069266634, 35146231883122, 1220316225699786, 45099576850647106, 1766204747811044266, 73012853049788936050, 3175271515525778312810

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=10 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 10], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 *)

A141008 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=12.

Original entry on oeis.org

1, 6, 64, 930, 16780, 356922, 8681380, 236739354, 7135453180, 235079261562, 8389974421012, 322019941283514, 13211434169884204, 576435716324437722, 26631611751773999044, 1297943388416061780186

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=12 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 12], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 *)

A141009 E.g.f. exp(Sum_{d|M} (exp(d*x)-1)/d), M=13.

Original entry on oeis.org

1, 2, 18, 262, 4498, 88174, 1989162, 51366438, 1491069602, 47749828830, 1664928894170, 62693869629142, 2534737217687378, 109469680507411214, 5025930552213949450, 244236790780300327302, 12515419830686362586882

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Also the number of partitions of {1..(13n)} that are invariant under a permutation consisting of n 13-cycles. - Danny Rorabaugh, Oct 29 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Column 13 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 13], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012, after Wouter Meeussen in similar sequences *)
mx = 16; p = 13; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)

A141010 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=14.

Original entry on oeis.org

1, 4, 40, 602, 11384, 253698, 6495616, 187940402, 6055160152, 214444348674, 8261055193008, 343357357970866, 15298169514407880, 726752780487465922, 36642712015230282784, 1952976182776961017138

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=14 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i - 1)&/@Divisors[j]), {i, k}]; Table[u[n, 14], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 *)

A141011 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=15.

Original entry on oeis.org

1, 4, 40, 612, 11976, 276836, 7336248, 219610532, 7331824360, 269576919908, 10796426089880, 466904884431268, 21658080409800264, 1071807203930733668, 56328256034260866296, 3131089417758323092388

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=15 of A162663.

u[0, j_]: = 1; u[k_, j_]: = u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 15], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 *)

A162664 a(n) = sigma(n) + tau(n)^2.

Original entry on oeis.org

2, 7, 8, 16, 10, 28, 12, 31, 22, 34, 16, 64, 18, 40, 40, 56, 22, 75, 24, 78, 48, 52, 28, 124, 40, 58, 56, 92, 34, 136, 36, 99, 64, 70, 64, 172, 42, 76, 72, 154, 46, 160, 48, 120, 114, 88, 52, 224, 66, 129, 88, 134, 58, 184, 88, 184, 96, 106, 64, 312, 66, 112, 140, 176, 100

Offset: 1

Franklin T. Adams-Watters, Jul 09 2009

nonn

Row 2 of A162663.

Shawn A. Broyles, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A035116, A162663.

Table[DivisorSigma[1, n] + DivisorSigma[0, n]^2, {n, 65}] (* Robert A. Russell, Apr 28 2018 *)

a(n) = sigma(n) + numdiv(n)^2; \\ Michel Marcus, Apr 28 2018

A162665 a(n) = sigma_2(n) + 3 sigma(n) tau(n) + tau(n)^3.

Original entry on oeis.org

5, 31, 42, 111, 70, 258, 106, 329, 235, 410, 202, 930, 262, 602, 612, 931, 406, 1373, 490, 1518, 948, 1106, 682, 2802, 957, 1418, 1364, 2274, 1030, 3540, 1162, 2715, 1860, 2162, 1940, 5097, 1606, 2594, 2436, 4882, 1942, 5316, 2122, 4290, 3986, 3578, 2506

Offset: 1

Franklin T. Adams-Watters, Jul 09 2009

Row 3 of A162663.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001157, A162663.

Table[DivisorSigma[2,n]+3DivisorSigma[1,n]DivisorSigma[0,n]+ DivisorSigma[ 0,n]^3,{n,50}] (* Harvey P. Dale, Jun 22 2013 *)

a(n) = my(f = factor(n), d = numdiv(f)); sigma(f, 2) + 3 * d * sigma(f) + d^3; \\ Amiram Eldar, Jan 02 2025

cp's OEIS Frontend

A036081 The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.

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A141005 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=8.

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A141006 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=9.

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A141007 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=10.

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A141008 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=12.

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A141009 E.g.f. exp(Sum_{d|M} (exp(d*x)-1)/d), M=13.

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A141010 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=14.

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A141011 E.g.f. exp(sum_{d|M} (exp(d*x)-1)/d), M=15.

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A162664 a(n) = sigma(n) + tau(n)^2.

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A162665 a(n) = sigma_2(n) + 3 sigma(n) tau(n) + tau(n)^3.