A088311 -id:A088311 - cp's OEIS frontend

A294467 Binomial transform of A088311.

1, 2, 5, 22, 113, 746, 6037, 55070, 548417, 6281938, 79935941, 1087584422, 16109401585, 255667890362, 4358283982613, 79893373511086, 1542859916102657, 31322024816838050, 676027617881188357, 15287136167625123638, 362322855217463741681

Offset: 0

Vaclav Kotesovec, Oct 31 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..443

Cf. A266232, A294467, A294468.

Cf. A218481, A294466, A281425, A095051.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(&*[1 + x^k: k in [1..50]]))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 15 2018

Table[Sum[Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]

x='x+O('x^30); Vec(serlaplace(exp(x)*eta(x^2)/eta(x))) \\ G. C. Greubel, Oct 15 2018

a(n) ~ exp(1) * n! * A000009(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(n/3) - n + 1) * n^(n - 1/4) / (4*3^(1/4)).
E.g.f.: exp(x) * Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Oct 15 2018

A294468 Inverse binomial transform of A088311.

Original entry on oeis.org

1, 0, 1, 8, 9, 224, 1225, 11304, 103537, 1431296, 15642801, 206721800, 3295533241, 47467875168, 859354139449, 15596241280424, 283240963555425, 5859309797252864, 129874369387025377, 2752905169704533256, 67640333903657850601

Offset: 0

Vaclav Kotesovec, Oct 31 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..443

Cf. A266232, A294467, A293467.

Cf. A218481, A294466, A281425, A095051.

Table[Sum[(-1)^(n-k)*Binomial[n, k]*k!*PartitionsQ[k], {k, 0, n}], {n, 0, 20}]
max = 20; t = Table[k!*PartitionsQ[k], {k, 0, max}]; Table[Differences[t, n], {n, 0, max}][[All, 1]] (* Jean-François Alcover, Nov 02 2017 *)

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A088311(k).
a(n) ~ exp(-1) * n! * A000009(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(n/3) - n - 1) * n^(n - 1/4) / (4*3^(1/4)).
E.g.f.: exp(-x) * Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Oct 15 2018

A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 6, 0, 24, 24, 0, 120, 240, 0, 720, 1440, 720, 0, 5040, 15120, 5040, 0, 40320, 120960, 80640, 0, 362880, 1451520, 1088640, 0, 3628800, 14515200, 14515200, 3628800, 0, 39916800, 199584000, 199584000, 39916800, 0, 479001600, 2395008000, 3353011200, 958003200

Offset: 0

Geoffrey Critzer, Feb 23 2022

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2;
  0,     6,      6;
  0,    24,     24;
  0,   120,    240;
  0,   720,   1440,   720;
  0,  5040,  15120,  5040;
  0, 40320, 120960, 80640;
  ...

Columns k=0-1 give: A000007, A000142 (for n>=1).
Cf. A088311 (row sums).
T(A000217(n),n) gives A052295.
Cf. A003056, A293140.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand(x*b(n-i, min(i-1, n-i)))*n!/(n-i)!))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 23 2022

nn = 13; Prepend[Map[Prepend[#, 0] &, Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Product[1 + y x^i, {i, 1, nn}], {x, 0, nn}],{x,y}]], 1]], {1}] // Grid

E.g.f.: Product_{i>=1} (1 + y*x^i).

Sum_{k=0..A003056(n)} (-1)^k * T(n,k) = A293140(n). - Alois P. Heinz, Feb 23 2022

A305550 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k).

Original entry on oeis.org

1, 1, 3, 19, 135, 1171, 12543, 156619, 2185095, 33787171, 579341583, 10927420219, 223956672855, 4940901389971, 116678668726623, 2938719256363819, 78709685812037415, 2234633592020685571, 67005923560416063663, 2114549937496479803419, 70024572874029038582775, 2427790107567416812409971

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A088311.
From Peter Bala, Jul 08 2022: (Start)
Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 1, 3, 3, 7, 3, 15, 11, 7, 3, 15, 11, 7, 3, 15, 11, ...], with an apparent period of 4 beginning at a(4). Cf. A167137.
More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000009, A088311, A167137, A306045, A320350.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(Stirling2(n, k)*k!*b(k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 21; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^k (Exp[x] - 1)^k/(k ((Exp[x] - 1)^k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} (-1)^k*(exp(x) - 1)^k/(k*((exp(x) - 1)^k - 1))).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A088311(k).
From Vaclav Kotesovec, Jun 17 2018: (Start)
a(n) ~ n! * exp(Pi*sqrt(n/(6*log(2))) + (1/log(2) - 1) * Pi^2/48) / (2^(9/4) * 3^(1/4) * n^(3/4) * (log(2))^(n + 1/4)).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(6*log(2))) + (1/log(2) - 1) * Pi^2/48 - n) * n^(n + 1/2) / (2^(7/4) * 3^(1/4) * n^(3/4) * (log(2))^(n + 1/4)).
(End)

A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 12, 0, 1, 1, 3, 12, 48, 0, 1, 1, 3, 13, 72, 360, 0, 1, 1, 3, 13, 72, 480, 2880, 0, 1, 1, 3, 13, 73, 500, 3780, 25200, 0, 1, 1, 3, 13, 73, 500, 4020, 35280, 241920, 0, 1, 1, 3, 13, 73, 501, 4050, 37380, 372960, 2903040, 0

Offset: 0

Seiichi Manyama, Oct 01 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   2,   3,   3,   3, ...
   0,  12,  12,  13,  13, ...
   0,  48,  72,  72,  73, ...
   0, 360, 480, 500, 500, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A088311, A293138, A293195, A293196, A293197.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293139.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
A:= (n, k)-> n!*b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Oct 02 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1, k]/j!, {j, 0, Min[k, n/i]}]]];
A[n_, k_] := n! b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

Original entry on oeis.org

1, 1, 4, 30, 312, 3720, 64080, 1305360, 33949440, 1019692800, 36360576000, 1487539468800, 69633899596800, 3649476307276800, 213929162589542400, 13848506938506240000, 986705192227442688000, 76724136092268048384000, 6491471142159880740864000

Offset: 0

Chad Brewbaker, Mar 27 2014

nonn

From Paul Boddington, Feb 24 2015: (Start)
Suppose G is the symmetric group on n letters. For each g in G, the set of f satisfying fgg = ggf is just the centralizer Z_gg(G). However |Z_gg(G)| is clearly constant on conjugacy classes of G. By the orbit-stabilizer theorem the size of the conjugacy class containing g is |G| / |Z_g(G)|. Since |G| = n! and Z_g(G) is a subgroup of Z_gg(G) we see that a(n) equals n! multiplied by the sum of indices |Z_gg(G) : Z_g(G)| where the sum is over representatives of the conjugacy classes of G. Since the conjugacy classes of G correspond to partitions of n (A000041), this makes it relatively easy to find terms.
a(n) appears to equal n! * A082733(n).
(End)

John F. Humphreys, A Course In Group Theory, Oxford Science Publications, 1996, chapter 10.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..60
Eric Weisstein's World of Mathematics, Symmetric Group

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

a(8)-a(9) from Giovanni Resta, Mar 27 2014

More terms from Paul Boddington, Feb 23 2015

A239836 Number of ordered pairs of permutation functions f,g on a size n set where f(g(g(x))) = g(f(f(x))).

Original entry on oeis.org

1, 1, 2, 6, 48, 360, 2880, 20160, 241920, 3265920, 47174400, 678585600, 12933043200, 193037644800, 3661488230400, 74537438976000, 1736591560704000, 36991492521984000

Offset: 0

Chad Brewbaker, Mar 27 2014

Cf. A000012, A000085, A000142, A088311, A053529, A001044, A239779, A255525.

a(n) = A255525(n) * n!.

a(8)-a(9) from Giovanni Resta, Mar 27 2014

a(10)-a(16) from Max Alekseyev, Jan 29 2025

A320350 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).

Original entry on oeis.org

1, 1, 3, 20, 148, 1384, 15728, 207696, 3094152, 51423288, 945943512, 19083180192, 418550811600, 9907493349168, 251588827187280, 6820899616891008, 196645361557479552, 6007407711127690752, 193842462200078260224, 6586904673329133618432, 235079477736802622742528, 8790132360155070084076800

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

Cf. A000009, A048994, A088311, A298905, A305550, A320349.

seq(n!*coeff(series(mul((1 + log(1/(1 - x))^k),k=1..100),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 21; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000009(k)*k!.
From Vaclav Kotesovec, Oct 13 2018: (Start)
a(n) ~ n! * exp(n + Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) / (4 * 3^(1/4) * n^(3/4) * (exp(1) - 1)^(n + 1/4)).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) * n^(n - 1/4) / (2^(3/2) * 3^(1/4) * (exp(1) - 1)^(n + 1/4)).
(End)

A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).

Original entry on oeis.org

1, 1, 1, 8, -8, 224, -712, 9120, -53496, 980088, -14394648, 264140832, -4113747024, 59028225840, -545558201424, -4191307074432, 450100910950272, -17302659472138752, 530508727766191104, -14790496500550616832, 408513443917280375808, -12274212131738107257600

Offset: 0

Ilya Gutkovskiy, Jun 18 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000009, A088311, A305550, A306023, A306042, A320350.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(Stirling1(n, j)*b(j)*j!, j=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 18 2018

nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} (-1)^(k+1)*log(1 + x)^k/(k*(1 - log(1 + x)^k))).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000009(k)*k!.

A330388 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * log(1 + x)^k / (k * (1 - log(1 + x)^k)).

Original entry on oeis.org

1, 0, 7, -37, 338, -2816, 28418, -340334, 5015080, -84244704, 1536606168, -29753884392, 609895549872, -13243687082016, 305507366834832, -7523621131117296, 198844500026698752, -5649686902983730560, 171839087043420258432, -5545292300345590210944

Offset: 1

Ilya Gutkovskiy, Dec 12 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A000009, A000593, A008275, A088311, A089064, A265024, A298905, A330354, A330387.

nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + Log[1 + x]^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^(2*k - 1)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A298905.
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000009.
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * A000593(k).
E.g.f.: Sum_{k>=1} log(1 + log(1 + x)^k). - Vaclav Kotesovec, Dec 15 2019
Conjecture: a(n) ~ n! * (-1)^(n+1) * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

cp's OEIS Frontend

A294467 Binomial transform of A088311.

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A294468 Inverse binomial transform of A088311.

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A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

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A305550 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k).

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A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

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A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

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A239836 Number of ordered pairs of permutation functions f,g on a size n set where f(g(g(x))) = g(f(f(x))).

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A320350 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).

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A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).

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