A088009 -id:A088009 - cp's OEIS frontend

A318976 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(phi(k)/k), where phi is the Euler totient function A000010.

1, 2, 6, 32, 196, 1512, 13384, 135872, 1545744, 19441952, 268386784, 4018603008, 65021744704, 1127284876928, 20880206388864, 410781080941568, 8561002328678656, 188224613741879808, 4355496092560324096, 105752112730661347328, 2688539359466319184896

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A088009 and A000262.

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

Cf. A000262, A088009, A318814, A318977.

Cf. A318975, A301555, A301554.

nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[E^(x*(2 + x)/(1 - x^2)), {x, 0, nmax}], x] * Range[0, nmax]!

E.g.f.: exp(x*(2 + x)/(1 - x^2)).

a(n) ~ 2^(-3/4) * 3^(1/4) * exp(sqrt(6*n) - n - 1/2) * n^(n - 1/4).

A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 481, 3841, 38221, 464353, 5368609, 82042561, 1151767981, 20242097041, 342921513793, 6705416722369, 133590317946541, 2880298682358721, 65597610230669761, 1556262483879791233, 39569880403136366029, 1030778206965403668721

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A012150, A088009, A373619.

Cf. A082579, A373578.

A373620 := proc(n)
    add(binomial(2*n-3*k-1,k)/(n-2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A373620(n),n=0..80) ; # R. J. Mathar, Jun 11 2024

a(n) = n!*sum(k=0, n\2, binomial(2*n-3*k-1, k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k)/(n-2*k)!.
a(n) == 1 mod 12.
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/48 + 2^(-5/3)*n^(1/3) + 3*2^(-4/3)*n^(2/3) - n) * n^(n - 1/6). - Vaclav Kotesovec, Jun 11 2024
D-finite with recurrence a(n) -a(n-1) -3*(n-1)*(n-2)*a(n-2) -3*(n-1)*(n-2)*a(n-3) +3*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jun 11 2024

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

Original entry on oeis.org

1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000726, A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := x*(1+x)/(1-x^3) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(0=
      irem(j, 3), 0, a(n-j)*j!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

CoefficientList[Series[E^(x*(1+x)/(1-x^3)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: exp(x*(1+x)/(1-x^3)).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 6^(-1/4) * n^(n-1/4) * exp(2/3*sqrt(6*n)-n) * (1 - 43/(48*sqrt(6*n))). - Vaclav Kotesovec, Sep 25 2013

2 more terms from R. J. Mathar, Feb 06 2008

A117347 Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

Original entry on oeis.org

4, 6, 8, 10, 16, 20, 28, 32, 64, 70, 88, 104, 110, 120, 128, 136, 152, 256, 464, 496, 512, 592, 650, 672, 884, 1024, 1155, 1888, 1952, 2048, 2144, 4030, 4096, 5830, 8128, 8192, 8384, 8925, 11096, 16384, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32768

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near". E.g., is sigma(n) (where sigma is the sum-of-divisors function) really "near" a multiple of n, for n = 9? Or n = 18?

			70 is a term because sigma(70) = 144 = 2 * 70 + 4, while 4 < log(70) ~= 4.248.

R. K. Guy, Unsolved Problems in Number Theory, B2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Nissen, Near Multiperfects.
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088009, A088010, A088011, A088012.
Cf. A117346, A117348, A117349, A117350.

sigma(m) = k * m + r, abs(r) <= log(m).

Offset corrected by Amiram Eldar, Mar 05 2020

A293486 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(2i-1))/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 6, 0, 1, 1, 1, 6, 24, 0, 1, 1, 1, 7, 24, 120, 0, 1, 1, 1, 7, 24, 180, 720, 0, 1, 1, 1, 7, 25, 180, 1080, 5040, 0, 1, 1, 1, 7, 25, 180, 1200, 10080, 80640, 0, 1, 1, 1, 7, 25, 181, 1200, 10080, 90720, 725760, 0, 1, 1, 1, 7, 25

Offset: 0

Seiichi Manyama, Oct 10 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   0,   1,   1,   1, ...
   0,   6,   6,   7,   7, ...
   0,  24,  24,  24,  25, ...
   0, 120, 180, 180, 180, ...

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

Columns k=0..3 give A000007, A293487, A293488, A293489.
Rows n=0 gives A000012.
Main diagonal gives A088009.
Cf. A293135.

A096965 Number of sets of even number of even lists, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 37, 241, 2101, 18271, 201097, 2270017, 29668681, 410815351, 6238931821, 101560835377, 1765092183037, 32838929702671, 644215775792401, 13441862819232001, 293976795292186897, 6788407001443004647, 163735077313046119861, 4142654439686285737201

Offset: 0

Vladeta Jovovic, Aug 18 2004

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

Cf. A000262, A088026, A088009, A096939, A349776.

a:= proc(n) option remember;  `if`(n<4, [1$3, 7][n+1], ((2*n-3)
      *a(n-1)+(n-1)*(2*n^2-8*n+7)*a(n-2) + (n-2)*(n-1)*(2*n-5)
      *a(n-3)-(n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4))/(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 01 2021

Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))]Cosh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)

E.g.f.: exp(x/(1-x^2))*cosh(x^2/(1-x^2)).
a(n) = (n!*sum(m=floor((n+1)/2)..n, (binomial(n-1,2*m-n-1))/(2*m-n)!)). - Vladimir Kruchinin, Mar 10 2013
Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
From Alois P. Heinz, Dec 01 2021: (Start)
a(n) = A000262(n) - A096939(n).
a(n) = |Sum_{k=0..n} (-1)^k * A349776(n,k)|. (End)

More terms from Robert G. Wilson v, Aug 19 2004

a(0)=1 prepended by Alois P. Heinz, Dec 01 2021

A351933 Expansion of e.g.f. exp(x / (1 - x^2/2)).

Original entry on oeis.org

1, 1, 1, 4, 13, 61, 331, 1996, 14449, 109873, 971821, 8995636, 93329941, 1018571269, 12110589583, 151955795356, 2037757374241, 28837620752161, 430834395468889, 6777014821152868, 111663525724783741, 1930478057636642221, 34781068833200111731

Offset: 0

Seiichi Manyama, Feb 26 2022

nonn

Cf. A000262, A351934, A351935, A351936.

Cf. A088009.

m = 22; Range[0, m]! * CoefficientList[Series[Exp[x / (1 - x^2/2)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^2/2))))

a(n) = if(n<3, 1, sum(k=0, (n-1)\2, (2*k+1)!/2^k*binomial(n-1, 2*k)*a(n-1-2*k)));

a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1)!/2^k * binomial(n-1,2*k) * a(n-1-2*k) for n > 2.
a(n) ~ n^(n - 1/4) / (2^(n/2 + 5/8) * exp(n - 2^(3/4)*sqrt(n))). - Vaclav Kotesovec, Mar 03 2022
Conjecture D-finite with recurrence +4*a(n) -4*a(n-1) -4*(n-1)*(n-2)*a(n-2) -2*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 09 2022
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k-1,k)/(2^k * (n-2*k)!). - Seiichi Manyama, Jun 08 2024

A373619 Expansion of e.g.f. exp(x / (1 - x^2)^(3/2)).

Original entry on oeis.org

1, 1, 1, 10, 37, 316, 2341, 21736, 237385, 2611792, 35911081, 476570656, 7654975021, 121021831360, 2196593121997, 40464132512896, 817485662059921, 17159299818547456, 382733978898335185, 8982388245979044352, 219867829220866999861, 5684505550914409716736

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A012150, A088009, A373620.

Cf. A373577.

a(n) = n!*sum(k=0, n\2, binomial(3*n/2-2*k-1, k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*n/2-2*k-1,k)/(n-2*k)!.
a(n) == 1 mod 9.
a(n) ~ 3^(1/5) * 5^(-1/2) * exp(3^(-1/5)*n^(1/5)/4 + 5*3^(-3/5)*n^(3/5)/2 - n) * n^(n - 1/5) * (1 - 1/(10*3^(4/5)*n^(1/5))). - Vaclav Kotesovec, Jun 11 2024

A373683 Expansion of e.g.f. exp(x / (1 - x^2)) / (1 - x^2).

Original entry on oeis.org

1, 1, 3, 13, 61, 441, 3031, 28813, 267513, 3088081, 36278731, 491262861, 6962025973, 108395586313, 1791145742751, 31601369155021, 594291393830641, 11740929829286433, 246910933786777363, 5406641472165854221, 125497950720670828461, 3018786042678264770521

Offset: 0

Seiichi Manyama, Jun 13 2024

nonn

Cf. A373681, A373682.

Cf. A088009.

a(n) = n!*sum(k=0, n\2, binomial(n-k, k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k,k)/(n-2*k)!.

a(n) == 1 (mod 2).

A096939 Number of sets of odd number of even lists, cf. A000262.

Original entry on oeis.org

0, 2, 6, 36, 260, 1950, 19362, 193256, 2326536, 29272410, 413257790, 6231230412, 101415565836, 1769925341366, 32734873484250, 646218442877520, 13404753632014352, 294656673023216946, 6775966692145553526

Offset: 1

Vladeta Jovovic, Aug 18 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000262, A088026, A088009, A096965.

Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))] Sinh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)

E.g.f.: exp(x/(1-x^2))*sinh(x^2/(1-x^2)).
Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = A000262(n) - A096965(n). - Alois P. Heinz, Dec 01 2021

More terms from Robert G. Wilson v, Aug 19 2004

cp's OEIS Frontend

A318976 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(phi(k)/k), where phi is the Euler totient function A000010.

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A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

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A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

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A117347 Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

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A293486 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*(2*i-1))/j!.

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A096965 Number of sets of even number of even lists, cf. A000262.

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Maple

Mathematica

Formula

Extensions

A351933 Expansion of e.g.f. exp(x / (1 - x^2/2)).

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A373619 Expansion of e.g.f. exp(x / (1 - x^2)^(3/2)).

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A373683 Expansion of e.g.f. exp(x / (1 - x^2)) / (1 - x^2).

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A293486 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(2i-1))/j!.