A059376 -id:A059376 - cp's OEIS frontend

A238743 Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).

1, 1, 1, 1, 7, 1, 1, 26, 26, 1, 1, 56, 208, 56, 1, 1, 124, 992, 992, 124, 1, 1, 182, 3224, 6944, 3224, 182, 1, 1, 342, 8892, 42408, 42408, 8892, 342, 1, 1, 448, 21888, 153216, 339264, 153216, 21888, 448, 1, 1, 702, 44928, 590976, 1920672, 1920672, 590976, 44928

Offset: 0

Tom Edgar, Mar 04 2014

We assume that A059382(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_3 given in A059376.
Another name might be the 3-totienomial coefficients.

			The first five terms in the third Jordan totient function are 1,7,26,56,124 and so T(4,2) = 56*26*7*1/((7*1)*(7*1))=208 and T(5,3) = 124*56*26*7*1/((26*7*1)*(7*1))=992.
The triangle begins
1
1 1
1 7   1
1 26  26   1
1 56  208  56   1
1 124 992  992  124  1
1 182 3224 6944 3224 182 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A059382, A059376, A238688, A238453.

q=100 #change q for more rows
P=[0]+[i^3*prod([1-1/p^3 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059382(n)/(A059382(k)* A059382(n-k)).
T(n,k) = prod_{i=1..n} A059376(i)/(prod_{i=1..k} A059376(i)*prod_{i=1..n-k} A059376(i)).
T(n,k) = A059376(n)/n*(k/A059376(k)*T(n-1,k-1)+(n-k)/A059376(n-k)*T(n-1,k)).

A309335 a(n) = n^3 if n odd, 7*n^3/8 if n even.

Original entry on oeis.org

0, 1, 7, 27, 56, 125, 189, 343, 448, 729, 875, 1331, 1512, 2197, 2401, 3375, 3584, 4913, 5103, 6859, 7000, 9261, 9317, 12167, 12096, 15625, 15379, 19683, 19208, 24389, 23625, 29791, 28672, 35937, 34391, 42875, 40824, 50653, 48013, 59319, 56000, 68921, 64827, 79507, 74536, 91125

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A007331.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A000578, A007331, A016755, A026741, A059376, A248882, A303383 (partial sums), A308422, A309336.

a[n_] := If[OddQ[n], n^3, 7 n^3/8]; Table[a[n], {n, 0, 45}]
nmax = 45; CoefficientList[Series[x (1 + 7 x + 23 x^2 + 28 x^3 + 23 x^4 + 7 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 7, 27, 56, 125, 189, 343}, 46]
Table[n^3 (15 - (-1)^n)/16, {n, 0, 45}]

G.f.: x * (1 + 7*x + 23*x^2 + 28*x^3 + 23*x^4 + 7*x^5 + x^6)/(1 - x^2)^4.
G.f.: Sum_{k>=1} J_3(k) * x^k/(1 - x^(2*k)), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) * (1 - 1/2^s).
a(n) = n^3 * (15 - (-1)^n)/16.
a(n) = Sum_{d|n, n/d odd} J_3(d).
Sum_{n>=1} 1/a(n) = 57*zeta(3)/56 = 1.223522205001729897639...
Multiplicative with a(2^e) = 7*2^(3*e-3), and a(p^e) = p^(3*e) for odd primes p. - Amiram Eldar, Oct 26 2020
Euler transform is A248882. - Georg Fischer, Nov 10 2020

A338548 a(n) = n^3 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^3.

Original entry on oeis.org

1, -9, 26, -56, 124, -234, 342, -448, 702, -1116, 1330, -1456, 2196, -3078, 3224, -3584, 4912, -6318, 6858, -6944, 8892, -11970, 12166, -11648, 15500, -19764, 18954, -19152, 24388, -29016, 29790, -28672, 34580, -44208, 42408, -39312, 50652, -61722, 57096, -55552, 68920, -80028

Offset: 1

Ilya Gutkovskiy, Nov 02 2020

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

Cf. A008683, A059376, A138503, A325596, A338547, A338549.

Table[n^3 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^3, {d, Divisors[n]}], {n, 1, 42}]
nmax = 42; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 4 x^k + x^(2 k))/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^3 - 1)*p^(3*(e - 1)); f[2, 1] = -9; f[2, e_] := -7*2^(3*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = n^3 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^3); \\ Michel Marcus, Nov 02 2020

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 4*x^k + x^(2*k)) / (1 + x^k)^4.
G.f. A(x) satisfies: A(x) = x * (1 - 4*x + x^2) / (1 + x)^4 - Sum_{k>=2} A(x^k).
Dirichlet g.f.: (1 - 2^(4 - s)) * zeta(s - 3) / zeta(s).
a(n) = J_3(n) if n odd, J_3(n) - 16 * J_3(n/2) if n even, where J_3 = A059376 (Jordan function J_3).
Multiplicative with a(2) = -9, a(2^e) = -7*2^(3*(e-1)) for e > 1, and a(p^e) = (p^3-1)*p^(3*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).

Original entry on oeis.org

1, 22, 105, 414, 745, 2310, 2737, 7134, 9231, 16390, 15961, 43470, 30745, 60214, 78225, 118238, 88417, 203082, 137161, 308430, 287385, 351142, 291985, 749070, 481245, 676390, 767391, 1133118, 731641, 1720950, 953281, 1924574, 1675905, 1945174, 2039065, 3821634

Offset: 1

Seiichi Manyama, May 26 2024

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

Cf. A059376, A328259, A372952.
Cf. A373131, A373133.
Cf. A013661, A013663.

f[p_, e_] := (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=1) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n) )^3 ).
a(n) = Sum_{d|n} J_3(d) * sigma(d), where the Jordan totient function J_3(n) = A059376(n)
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 1/p^4 - 1/p^5 + 1/p^6) = 1.54488120152452251241... . (End)

A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^2 ).

Original entry on oeis.org

1, 50, 339, 1786, 3845, 16950, 19495, 58682, 85281, 192250, 176891, 605454, 401869, 974750, 1303455, 1890106, 1507985, 4264050, 2612899, 6867170, 6608805, 8844550, 6727799, 19893198, 12109345, 20093450, 20802003, 34818070, 21241949, 65172750, 29581471, 60581690

Offset: 1

Seiichi Manyama, May 26 2024

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

Cf. A059376, A372962.
Cf. A373130, A373133.
Cf. A013664.

f[p_, e_] := (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3 ).
a(n) = Sum_{d|n} J_3(d) * sigma(d^2), where the Jordan totient function J_3(n) = A059376(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4) = 1.67666099579383196077... . (End)

A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

Original entry on oeis.org

1, 3, 5, 13, 7, 19, 9, 43, 18, 33, 13, 93, 15, 51, 35, 137, 19, 110, 21, 175, 45, 99, 25, 355, 38, 129, 58, 285, 31, 289, 33, 455, 65, 201, 63, 626, 39, 243, 75, 721, 43, 483, 45, 589, 126, 339, 49, 1305, 66, 498, 95, 783, 55, 750, 91, 1227

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..20000
G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.

Cf. A263825-A263830, A263832.

A001001(n) = sumdiv(n, d, sigma(d) * d);
A007429(n) = sumdiv(n, d, sigma(d));
A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d,2));
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
S22(n)     = { if (n%2, A060640(n), if (n%4, 0,
  sumdiv(n\4, d, 2*d*(sigma(n\(2*d)) - sigma(n\(4*d))))));
};
A027844(n) = S1(n) + S11(n) + S21(n);
a(n) = { 1/n * sumdiv(n, d,
  A059376(d) * S1(n\d) + EpiPcZn(d) * S21(n\d) + A007434(d) * S22(n\d));
};
vector(56, n, a(n))  \\ Gheorghe Coserea, May 04 2016

More terms from Gheorghe Coserea, May 04 2016

A308418 Expansion of e.g.f. exp(x(1 + 3x + 6x^2 + 3x^3 + x^4)/(1 - x^2)^3).

Original entry on oeis.org

1, 1, 7, 73, 649, 8821, 122311, 2064637, 37933393, 773276329, 17257075111, 414876953041, 10780187135257, 298418920103773, 8812636845668839, 275368711393020421, 9091457478119636641, 315782978460465185617, 11511089733834178827463, 439231563093877354663129

Offset: 0

Ilya Gutkovskiy, May 25 2019

nonn

Cf. A007434, A059376, A088009, A255807, A308417, A308422.

nmax = 19; CoefficientList[Series[Exp[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Product[(1 + x^k)^(DirichletConvolve[j^3, MoebiusMu[j], j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[1/8 (7 - (-1)^k) k^2 k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

my(x ='x + O('x^30)); Vec(serlaplace(exp(x*(1+3*x+6*x^2+3*x^3+x^4)/(1-x^2)^3))) \\ Michel Marcus, May 26 2019

E.g.f.: exp(Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k))), where J_2() is the Jordan function (A007434).
E.g.f.: Product_{k>=1} (1 + x^k)^(J_3(k)/k), where J_3() is the Jordan function (A059376).
a(n) ~ 2^(-5/4) * 21^(1/8) * n^(n - 1/8) * exp(2^(3/2) * 3^(-3/4) * 7^(1/4) * n^(3/4) - n). - Vaclav Kotesovec, May 28 2019
E.g.f.: exp(Sum_{k>=1} A308422(k)*x^k). - Ilya Gutkovskiy, May 29 2019

A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

Original entry on oeis.org

1, 9, 27, 73, 125, 243, 343, 585, 729, 1125, 1331, 1971, 2197, 3087, 3375, 4681, 4913, 6561, 6859, 9125, 9261, 11979, 12167, 15795, 15625, 19773, 19683, 25039, 24389, 30375, 29791, 37449, 35937, 44217, 42875, 53217, 50653, 61731, 59319, 73125, 68921, 83349, 79507, 97163, 91125

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

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

Cf. A000578, A001511, A016755, A023872, A059376, A129527, A209229, A328407.

[n eq 1 select 1 else IsOdd(n) select n^3 else Self(n div 2)+n^3 :n in [1..45]]; // Marius A. Burtea, Oct 15 2019

nmax = 45; CoefficientList[Series[Sum[x^(2^k) (1 + 4 x^(2^k) + x^(2^(k + 1)))/(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n^3, n^3]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^3 &], {n, 1, 45}]
f[p_, e_] :=p^(3*e); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

G.f.: Sum_{k>=0} x^(2^k) * (1 + 4*x^(2^k) + x^(2^(k+1))) / (1 - x^(2^k))^4.
G.f.: (1/7) * Sum_{k>=1} J_3(2*k) * x^k / (1 - x^k), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) / (1 - 2^(-s)).
a(2*n) = a(n) + 8*n^3, a(2*n+1) = (2*n + 1)^3.
a(n) = Sum_{d|n} A209229(n/d) * d^3.
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023872.
Sum_{k=1..n} a(k) ~ 4*n^4/15. - Vaclav Kotesovec, Oct 15 2019
Multiplicative with a(2^e) = (8^(e+1)-1)/7, and a(p^e) = p^(3*e) for an odd prime p. - Amiram Eldar, Oct 23 2023

A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.

Original entry on oeis.org

1, 73, 757, 4672, 15751, 55261, 117993, 299008, 551853, 1149823, 1772893, 3536704, 4829007, 8613489, 11923507, 19136512, 24142483, 40285269, 47052741, 73588672, 89320701, 129421189, 148048057, 226349056, 246109375, 352517511, 402300837, 551263296

Offset: 1

Seiichi Manyama, Mar 05 2025

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A065958, A160889, A194532.
Cf. A059376, A069094.
Cf. A013664, A013667.

f[p_, e_] := p^(6*e) * (1 + 1/p^3 + 1/p^6); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 28] (* Amiram Eldar, Mar 05 2025 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n) = J(n, 9)/J(n, 3);

a(n) = {my(p = factor(n)[, 1]); n^6 * prod(i = 1, #p, 1 + 1/p[i]^3 + 1/p[i]^6);} \\ Amiram Eldar, Mar 05 2025

a(n) = A069094(n)/A059376(n).
a(n) = n^6 * Product_{distinct primes p dividing n} (1 + 1/p^3 + 1/p^6).
From Amiram Eldar, Mar 05 2025: (Start)
Dirichlet g.f.: zeta(s-6) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = Product_{p prime} (1 + 1/p^4 + 1/p^7) = 1.08635980686198102055... .
Sum_{n>=1} 1/a(n) = zeta(6)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^15) = 1.01533121878447451064... . (End)

A338165 Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

Original entry on oeis.org

1, 14, 52, 161, 248, 728, 684, 1680, 2080, 3472, 2660, 8372, 4392, 9576, 12896, 16576, 9824, 29120, 13716, 39928, 35568, 37240, 24332, 87360, 46376, 61488, 74412, 110124, 48776, 180544, 59580, 157696, 138320, 137536, 169632, 334880, 101304, 192024, 228384, 416640

Offset: 1

Ilya Gutkovskiy, Oct 14 2020

Dirichlet convolution of Jordan function J_3 (A059376) with itself.

Wikipedia, Jordan's totient function

Cf. A000005, A007427, A029935, A059376, A338164.

Jordan3[n_] := Sum[d^3 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan3[d] Jordan3[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]
a[1] = 1; f[p_, e_] := p^(3 e - 6) (p^6 + e (p^3 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 40}]

Multiplicative with a(p^e) = p^(3*e - 6) * (p^6 + e * (p^3 - 1)^2 - 1).
a(n) = Sum_{d|n} J_3(d) * J_3(n/d).
a(n) = Sum_{d|n} d^3 * tau(d) * A007427(n/d), where tau = A000005.
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^3.
Sum_{k=1..n} a(k) ~ 2025 * n^4 * ((log(n) + 2*gamma - 1/4)/Pi^8 - 180*zeta'(4) / Pi^12), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020

cp's OEIS Frontend

A238743 Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).

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A309335 a(n) = n^3 if n odd, 7*n^3/8 if n even.

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A338548 a(n) = n^3 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^3.

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A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).

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A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^2 ).

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A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

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A308418 Expansion of e.g.f. exp(x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3).

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A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

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A308418 Expansion of e.g.f. exp(x(1 + 3x + 6x^2 + 3x^3 + x^4)/(1 - x^2)^3).