A053529 -id:A053529 - cp's OEIS frontend

A323295 Number of ways to fill a matrix with the first n positive integers.

1, 1, 4, 12, 72, 240, 2880, 10080, 161280, 1088640, 14515200, 79833600, 2874009600, 12454041600, 348713164800, 5230697472000, 104613949440000, 711374856192000, 38414242234368000, 243290200817664000, 14597412049059840000, 204363768686837760000

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(4) = 72 matrices consist of:
  24 row/column permutations of [1 2 3 4]
+
  4 row/column permutations of [1 2]
                               [3 4]
+
  4 row/column permutations of [1 2]
                               [4 3]
+
  4 row/column permutations of [1 3]
                               [2 4]
+
  4 row/column permutations of [1 3]
                               [4 2]
+
  4 row/column permutations of [1 4]
                               [2 3]
+
  4 row/column permutations of [1 4]
                               [3 2]
+
  24 row/column permutations of [1]
                                [2]
                                [3]
                                [4]

Cf. A000005, A000142, A038041, A053529, A057625, A061095, A120733, A121860.

Cf. A323300, A323301, A323307, A323351.

Join[{1}, Table[DivisorSigma[0, n]*n!, {n, 30}]]

a(n) = if (n==0, 1, numdiv(n)*n!); \\ Michel Marcus, Jan 15 2019

a(n) = A000005(n) * n! for n > 0, a(0) = 1.

E.g.f.: 1 + Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 13 2019

A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 21, 25, 37, 53, 137, 153, 249, 337, 505, 845, 1085, 1497, 2061, 2785, 3661, 7589, 8849, 13329, 18033, 26017, 34225, 48773, 70805, 91977, 123765, 164761, 216373, 283205, 367913, 470889, 758793, 913825, 1264105, 1651613, 2251709, 2894793, 3927837

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(6) = 21 matrices:
  [6] [1 5] [5 1] [2 4] [4 2] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [2] [4]
  [5] [1] [4] [2]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000009, A000142, A053529, A294617, A321654, A323295, A323300, A323307, A323351.

b:= proc(n, i, t) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),
       b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Sum[Length[ptnmats[k]],{k,Select[Times@@Prime/@#&/@IntegerPartitions[n],SquareFreeQ]}],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,
     If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +
     b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

a(n) = Sum_{y1 + ... + yk = n, y1 > ... > yk} k! * A000005(k) for n > 0, a(0) = 1.

a(0)=1 prepended by Alois P. Heinz, Jan 15 2019

A346547 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(x).

Original entry on oeis.org

1, 1, 6, 36, 282, 2575, 28075, 340809, 4657996, 69874305, 1145441713, 20279904337, 386803154474, 7874727448757, 170678885319787, 3919163707551187, 95029714996046680, 2424604353738271201, 64940619086990938317, 1820746123923294245293, 53328181409328560026038

Offset: 0

Ilya Gutkovskiy, Sep 16 2021

nonn

Exponential transform of A002745.

Cf. A000203, A002745, A053529, A346545, A346546, A346548.

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002745[n_] := Sum[Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002745[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002745(k) * a(n-k).

A095051 E.g.f.: exp(-x)/eta(x), where eta(x) is the Dedekind eta function.

Original entry on oeis.org

1, 0, 3, 8, 69, 384, 4375, 34152, 464457, 5051456, 75865131, 1032865800, 18108977293, 286975230528, 5639956035519, 105513165321704, 2269311347406225, 48066460265622912, 1146324511845384787, 26924271371612501256, 701472699537610875861, 18214089447110112972800, 512194770431254272442983

Offset: 0

Benoit Cloitre, Jun 19 2004

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
N. J. A. Sloane, Transforms

Cf. A218481, A294466, A281425.

Cf. A266232, A294467, A293467, A294468.

Table[Sum[(-1)^(n-k) * Binomial[n, k] * k! * PartitionsP[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 31 2017 *)
nmax = 20; CoefficientList[Series[Exp[-x] * x^(1/24)/DedekindEta[Log[x]/(2*Pi*I)], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 31 2017 *)

PARI
```
a(n)=polcoeff(1/eta(x)/exp(x),n)*n!
```

Inverse binomial transform of A053529. - Vladeta Jovovic, Jun 21 2004
From Vaclav Kotesovec, Oct 31 2017: (Start)
a(n) ~ exp(-1) * n! * A000041(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(2*n/3) - n - 1) * n^(n - 1/2) / (4*sqrt(3)). (End)
E.g.f.: exp(Sum_{k>=2} sigma(k)*x^k/k). - Ilya Gutkovskiy, Oct 15 2018

More terms from Michel Marcus, Oct 31 2017

A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, 1, 7, 43, 409, 3841, 50431, 648187, 10347793, 170363809, 3200390551, 62855417131, 1371594161257, 31147757782753, 768384638386639, 19814802390611131, 545309251861956001, 15661899520801953217, 475833949719419469223, 15042718034104688144299

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 14 2017: (Start)
The terms of the sequence appear to be of the form 6*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), this sequence (k=1), A294362 (k=2).

Cf. A000203, A053529, A274804, A061256, A192065.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 04 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3)/2 - 3^(1/3) * n^(1/3) / (2*Pi^(2/3)) + 1/24 - 1/(8*Pi^2) - n) * n^(n - 1/6) / 3^(2/3). - Vaclav Kotesovec, Sep 04 2018

A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 8, 38, 262, 1732, 16144, 153596, 1660796, 19415384, 264084064, 3664187848, 57366995272, 936097392752, 16131362629568, 302946516251408, 6034409270818576, 125044362929875744, 2756094464546395264, 63280996793936902496

Offset: 0

Vaclav Kotesovec, Sep 05 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A088009, A318769, A318696.
Cf. A000262, A305127, A318695.
Cf. A053529, A028342.
Cf. A000010 (phi), A008683 (mu).

nmax = 20; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[EulerPhi[k]* a[n-k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 29 2022

a(n)/n! ~ 3^(1/4) * exp(2*sqrt(6*n)/Pi) / (Pi * 2^(3/4) * n^(3/4)).
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} mu(gcd(k,j)). - Ilya Gutkovskiy, Aug 17 2021
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 29 2022
E.g.f.: exp( Sum_{n>=1} (mu(n)/n) * x^n/(1 - x^n) ), where mu(n) = A008683(n). - Paul D. Hanna, Jun 24 2023

A320349 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

Original entry on oeis.org

1, 1, 5, 32, 278, 2894, 35986, 514128, 8306448, 149558688, 2968216944, 64314676128, 1510065781968, 38178537908016, 1033794746169168, 29840453678758272, 914461132860063360, 29645845798652997120, 1013511411165693991680, 36436289007997132646400, 1373976152501162688288000

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..413

Cf. A000041, A000203, A048994, A053529, A167137, A306042, A320350.

seq(n!*coeff(series(mul(1/(1-log(1/(1-x))^k),k=1..100),x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] PartitionsP[k] k!, {k, 0, n}], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1/(1 - x))^k/k).
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000041(k)*k!.
From Vaclav Kotesovec, Oct 13 2018: (Start)
a(n) ~ n! * exp(n + Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) / (4 * sqrt(3) * n * (exp(1) - 1)^n).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) * n^(n - 1/2) / (2^(3/2) * sqrt(3) * (exp(1) - 1)^n).
(End)

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a Young diagram with the prime indices of n such that all rows are weakly increasing and all columns are strictly increasing.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Is this a duplicate of A339887? - R. J. Mathar, Feb 03 2021

			The a(60) = 5 tableaux:
  1123
.
  11   112   113
  23   3     2
.
  11
  2
  3

Cf. A000085, A000219, A003293, A053529, A056239, A112798, A138178, A153452, A296188.

Cf. A323300, A323429, A323432, A323436, A323438, A323439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Sum_{A056239(n) = k} a(k) = A003293(n).

A179973 Number of permutations of [n] whose cycle lengths are nondecreasing when cycles are ordered by their minima and these minima are {1..k} (for some k <= n).

Original entry on oeis.org

1, 1, 2, 4, 12, 42, 216, 1200, 8664, 66384, 612264, 5910024, 66723384, 776642664, 10311400344, 141065450904, 2153769250584, 33743736435864, 583781959921944, 10308436641381144, 198863818304824344, 3914117125411211544, 83301822014343774744, 1805447764831655109144

Offset: 0

Alford Arnold, Aug 05 2010

nonn

The original name was: Row sums of A179972 and also of A179974.

			a(4) = 12 = 6 + 2 + 2 + 1 + 1: (1234), (1243), (1324), (1342), (1423), (1432),
  (13)(24), (14)(23), (1)(234), (1)(243), (1)(2)(34), (1)(2)(3)(4).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000041, A000142, A000522, A004086, A053529, A179972, A179974, A327711, A362362, A364128.

a:= n-> add((n-nops(p))!, p=combinat[partition](n)):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 09 2023
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
     (p-n)!, b(n, i-1, p)+b(n-i, min(n-i, i), p-1))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 09 2023

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p - n)!, b[n, i - 1, p] + b[n - i, Min[n - i, i], p - 1]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

From Alois P. Heinz, Jul 09 2023: (Start)
a(n) = Sum_{lambda in partitions(n)} (n - |lambda|)!.
Limit_{n->oo} A004086(a(n))/10^A055642(a(n)) = A364128. (End)

Edited by R. J. Mathar, May 17 2016

a(0), a(9)-a(23) and new name from Alois P. Heinz, Jul 09 2023

cp's OEIS Frontend

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A095051 E.g.f.: exp(-x)/eta(x), where eta(x) is the Dedekind eta function.

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A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

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A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

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A320349 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

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A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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