A061684 -id:A061684 - cp's OEIS frontend

A061698 Erroneous version of A061684.

1, 1, 5, 64, 1613, 69026, 4566992, 437665649, 57903766800

Offset: 0

dead

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4. See Table VII.

A091315 Number of orbits of length n under the map whose periodic points are counted by A061684.

1, 2, 21, 402, 13805, 761154, 62523664, 7237970648, 1132600004910, 231900134422880, 60528794385067778, 19713593779259862624, 7869483395065035685162, 3792402572391137423764584

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061684 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n.

			The sequence A061684 begins 1,1,5,64,1613, so a(3)=(b(3)-b(1))/3=21.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Thomas Ward, Exactly realizable sequences. [local copy].

Cf. A061684.

If b(n) is the (n+1)th term in A061684, then a(n) = (1/n)*Sum_{d|n}mu(d)b(n/d).

Name clarified by Michel Marcus, May 14 2015

A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 5, 16, 15, 1, 1, 1, 9, 64, 131, 52, 1, 1, 1, 17, 298, 1613, 1496, 203, 1, 1, 1, 33, 1540, 25097, 69026, 22482, 877, 1, 1, 1, 65, 8506, 461105, 4383626, 4566992, 426833, 4140, 1, 1, 1, 129, 48844, 9483041, 350813126, 1394519922, 437665649, 9934563, 21147, 1

Offset: 0

Alois P. Heinz, Jul 14 2016

			A(2,2) = 3: 1234, 12|34, 14|23.
A(2,3) = 5: 123456, 123|456, 126|345, 135|246, 156|234.
A(2,4) = 9: 12345678, 1234|5678, 1238|4567, 1247|3568, 1278|3456, 1346|2578, 1368|2457, 1467|2358, 1678|2345.
A(3,2) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34.
Square array A(n,k) begins:
  1,   1,     1,       1,          1,            1,               1, ...
  1,   1,     1,       1,          1,            1,               1, ...
  1,   2,     3,       5,          9,           17,              33, ...
  1,   5,    16,      64,        298,         1540,            8506, ...
  1,  15,   131,    1613,      25097,       461105,         9483041, ...
  1,  52,  1496,   69026,    4383626,    350813126,     33056715626, ...
  1, 203, 22482, 4566992, 1394519922, 573843627152, 293327384637282, ...

Alois P. Heinz, Antidiagonals n = 0..60, flattened
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000110, A023998, A061684, A061685, A061686, A061687, A061688, A275097, A275098, A275099.
Rows n=0+1,2-5 give: A000012, A094373, A275100, A275101, A275102.
Main diagonal gives A275044.
Cf. A345400.

A:= proc(n, k) option remember; `if`(k*n=0, 1, add(
       binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*A[j, k], {j, 0, n-1}]/n]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)

A061692 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 1, 4, 1, 27, 36, 1, 172, 864, 576, 1, 1125, 17500, 36000, 14400, 1, 7591, 351000, 1746000, 1944000, 518400, 1, 52479, 7197169, 80262000, 191394000, 133358400, 25401600, 1, 369580, 151633440, 3691514176, 17188416000, 23866214400, 11379916800, 1625702400

Offset: 1

N. J. A. Sloane, Jun 19 2001

			1; 1,4; 1,27,36; 1,172,864,576; ...

Alois P. Heinz, Rows n = 1..100, flattened
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Diagonals give A001044, A061695, A061693, A061694. Cf. A061691.

Row sums give A061684.

b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^3, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!^3):
seq(T(n), n=1..10);  # Alois P. Heinz, Sep 10 2019

R[0, ] = 1; R[n, x_] := R[n, x] = x*Sum[Binomial[n, k]^2*Binomial[n-1, k]*R[k, x], {k, 0, n-1}]; Table[CoefficientList[R[n, x], x] // Rest, {n, 1, 8}] // Flatten (* Jean-François Alcover, Sep 01 2015, after Peter Bala *)

T(n, k) = 1/k!*Sum multinomial(n, n_1, n_2, ..n_k)^3, where the sum extends over all compositions (n_1, n_2, .., n_k) of n into exactly k nonnegative parts. - Vladeta Jovovic, Apr 23 2003

The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*( sum {k = 0..n-1} binomial(n,k)^2*binomial(n-1,k)*R(k,x) ) with R(0,x) = 1. Also R(n,x + y) = sum {k = 0..n} binomial(n,k)^3*R(k,x)*R(n-k,y). - Peter Bala, Sep 17 2013

More terms from Vladeta Jovovic, Apr 23 2003

A342197 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^3 * a(k-1).

Original entry on oeis.org

1, 1, 9, 63, 919, 18919, 505639, 18602319, 877402487, 51212704151, 3688010412503, 321523601578079, 33283248550719793, 4050897039400696253, 574469890816237292037, 93943844587040615104177, 17565329004174205621822169, 3730161837629377369026433019

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A040027, A061684, A336195, A342196, A342198, A342199.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^3 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 17}]

A336210 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k).

Original entry on oeis.org

1, -1, 3, -10, -117, 5224, -23010, -10319891, 463834315, 69461529092, -10005601418172, -1323175060249241, 468450359815048182, 63281374513705043227, -46495538420749056681263, -7147072328212024308730535, 9119277358213513566069911755, 1827085356172328516064256064092

Offset: 0

Ilya Gutkovskiy, Jul 12 2020

sign

Cf. A000587, A061684, A336209.

a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^3 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[-Sum[x^k/(k!)^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3

a(n) = (n!)^3 * [x^n] exp(-Sum_{k>=1} x^k / (k!)^3).

A352466 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2n,2k)^3 * k * a(n-k).

Original entry on oeis.org

1, 1, 109, 124876, 704029453, 13294133177626, 665514245564815384, 75462508236267111825685, 17305487139219914670764064013, 7368678746697280907127091048286734, 5449131877967324738667220718996986592734, 6632563741264033978048120096103173533343094035

Offset: 0

Ilya Gutkovskiy, Mar 17 2022

nonn

Cf. A005046, A061684, A352465, A352469.

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[2 n, 2 k]^3 k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 11}]
nmax = 22; Take[CoefficientList[Series[Exp[Sum[x^(2 k)/(2 k)!^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3, {1, -1, 2}]

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^3 = exp( Sum_{n>=1} x^(2*n) / (2*n)!^3 ).

A061699 Generalized Bell numbers.

Original entry on oeis.org

1, 0, 1, 1, 109, 1001, 128876, 4682637, 792013069, 75022864345, 17347941915130, 3306782335589129, 1063995670771466456, 344173484059603653963, 153912612667103172679837, 75571251960991348967876564, 46271172080109731069460430093, 32072712892330804080630204907257

Offset: 0

N. J. A. Sloane, Jun 19 2001

nonn

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Cf. A061684, A061700.

Sum_{n>=0} a(n) * x^n / (n!)^3 = exp(Sum_{n>=2} x^n / (n!)^3). - Ilya Gutkovskiy, Jul 12 2020

More terms from Ilya Gutkovskiy, Jul 12 2020

A061700 Generalized Bell numbers.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 4001, 42876, 347117, 792865081, 37062990505, 1164982989754, 2135094241854476, 289654511654619255, 24938050464296749001, 41388115708273073076689, 12793631315199589229518093, 2452257460931091883072686073, 3961922987460317585057396895353

Offset: 0

N. J. A. Sloane, Jun 19 2001

nonn

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Cf. A061684, A061699.

Sum_{n>=0} a(n) * x^n / (n!)^3 = exp(Sum_{n>=3} x^n / (n!)^3). - Ilya Gutkovskiy, Jul 12 2020

More terms from Ilya Gutkovskiy, Jul 12 2020

A346185 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k).

Original entry on oeis.org

1, 1, 2, 11, 93, 1294, 26045, 714391, 26109426, 1224739755, 71807248783, 5173027197636, 450173748220033, 46617339568635115, 5677430539873463470, 804907754967314483801, 131598260940217897338131, 24609634809861999705338820, 5226508081059269450476666513

Offset: 0

Ilya Gutkovskiy, Jul 09 2021

nonn

Cf. A000110, A061684, A101514, A181543, A336195.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^3 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]

cp's OEIS Frontend

A061698 Erroneous version of A061684.

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A091315 Number of orbits of length n under the map whose periodic points are counted by A061684.

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A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

A061692 Triangle of generalized Stirling numbers.

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Maple

Mathematica

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A342197 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^3 * a(k-1).

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Mathematica

A336210 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k).

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A352466 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2*n,2*k)^3 * k * a(n-k).

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A061699 Generalized Bell numbers.

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A061700 Generalized Bell numbers.

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A346185 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k).

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A352466 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2n,2k)^3 * k * a(n-k).