A129373 -id:A129373 - cp's OEIS frontend

A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

1, 1, 2, 3, 6, 8, 15, 20, 35, 48, 76, 103, 166, 221, 333, 451, 671, 894, 1303, 1730, 2479, 3288, 4615, 6086, 8502, 11142, 15299, 20034, 27285, 35514, 47937, 62168, 83259, 107650, 142929, 184090, 243207, 312041, 409210, 523709, 683261, 871239, 1130703

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A129375.

{a(n)=local(A=1+x);for(i=2,n,A=1/(1-x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

G.f.: A(x) = Product_{n>=1} 1/(1 - x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ exp((1 + 1/r) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / ((2*Pi)^(29/50) * sqrt(1+r) * n^((6 + 5*r)/(10*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

Original entry on oeis.org

5, 5, 0, 0, 1, 0, 0, 0, 5, 4, 1, 3, 1, 5, 4, 4, 9, 1, 8, 3, 3, 0, 5, 8, 1, 2, 6, 7, 0, 2, 2, 2, 2, 1, 9, 6, 4, 6, 1, 1, 6, 8, 2, 2, 7, 1, 0, 2, 7, 1, 4, 0, 4, 0, 9, 8, 8, 8, 3, 9, 6, 5, 8, 5, 8, 9, 2, 9, 0, 5, 3, 0, 6, 6, 6, 6, 0, 5, 6, 4, 8, 5, 9, 5, 1, 1, 8, 7, 2, 0, 6, 5, 2, 3, 5, 3, 4, 6, 6, 5, 4

Offset: 0

Jean-François Alcover, Sep 22 2014

			0.55001000541315449183305812670222219646116822710271404...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 41.

Cf. A107311, A129373, A129374, A129375, A217598, A247668.

digits = 101; rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> digits+5]; cm = -1/Zeta'[rho]; RealDigits[cm, 10, digits] // First

c_m = -1/zeta'(rho), where rho = 1.728647... is A107311, the real solution to zeta(rho) = 2.

Residue_{s = rho} 1/(2 - Zeta(s)). - Vaclav Kotesovec, Nov 04 2018

A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

Original entry on oeis.org

1, 1, 3, 13, 97, 621, 6571, 58633, 779073, 9317017, 138628531, 1977676581, 37384244833, 620735382853, 12434855135067, 245117537189281, 5651550278494081, 123266430844431153, 3128700944169196003

Offset: 0

Paul D. Hanna, Apr 12 2007

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A074206, A129373, A129374.

{a(n)=local(A=1+x);for(i=2,n,A=exp(x+x*O(x^n))*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); n!*polcoeff(A,n)}

E.g.f.: A(x) = exp( Sum_{n>=1} A074206(n)*x^n ) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ n! * exp(2/5 + (1 + 1/r) * (-Gamma(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) / Zeta'(r))^(1/(2*(1+r))) / (sqrt(2*Pi*(1+r)) * n^((2+r)/(2*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

PARI program fixed by Vaclav Kotesovec, Feb 26 2014

A307605 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

Original entry on oeis.org

1, 1, 2, 5, 12, 20, 48, 81, 169, 305, 580, 1009, 1966, 3338, 6067, 10503, 18730, 31633, 55641, 93151, 160389, 267585, 452762, 747016, 1253644, 2049943, 3390786, 5516227, 9034745, 14572790, 23668066, 37918484, 61042425, 97231826, 155292944, 245774727, 389998116

Offset: 0

Ilya Gutkovskiy, Apr 18 2019

nonn

Weigh transform of A050369.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 20*x^5 + 48*x^6 + 81*x^7 + 169*x^8 + 305*x^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A050369, A074206, A129373, A307604, A307606.

terms = 36; A[] = 1; Do[A[x] = (1 + x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} (1 + x^k)^(k*A074206(k)).

a(n) ~ ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(4 + 2*r)) * exp((2+r)/(1+r) * ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (2^(1/50) * sqrt(Pi*(2+r)) * n^((3 + r)/(4 + 2*r))), where r = A107311 is the root of the equation zeta(r) = 2. - Vaclav Kotesovec, Mar 18 2021

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 6, 19, 30, 96, 152, 461, 775, 1883, 3271, 8751, 14370, 34004, 59491, 140450, 239746, 541817, 932681, 2089189, 3606641, 7719178, 13398411, 28848808, 49603982, 103047935, 179154858, 370200348, 639269735, 1295389370, 2241994088, 4511677298, 7798101800, 15408901600

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Euler transform of A163767.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A000219, A001970, A006171, A129373, A129374, A163767, A174465, A280487.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A163767(k).

A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 52, 88, 152, 252, 416, 664, 1076, 1684, 2636, 4060, 6248, 9444, 14292, 21312, 31748, 46796, 68804, 100200, 145784, 210240, 302520, 432428, 616716, 873972, 1236136, 1738560, 2439936, 3407924, 4749160, 6589156, 9123976, 12582620, 17316052, 23745756

Offset: 0

Seiichi Manyama, Nov 04 2018

nonn

Convolution of A129373 and A129374. - Vaclav Kotesovec, Nov 05 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A129374.

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A074206(k) where A074206(n) is the number of ordered factorizations of n.

a(n) ~ exp((1+r) * ((2^(1+r) - 1) * Gamma(1+r) * Zeta(1+r))^(1/(1+r)) * n^(r/(1+r)) / (r * 2^(r/(1+r)) * (-Zeta'(r))^(1/(1+r)))) * (-2*(2^(1+r) - 1) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / (2^(7/25) * Pi^(29/50) * sqrt(1+r) * n^((6+5*r)/(10*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 05 2018

A308272 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)A(x^3)A(x^5)* ... A(x^prime(k)) ...

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 13, 16, 22, 27, 33, 44, 53, 65, 84, 101, 124, 156, 187, 226, 280, 336, 403, 492, 587, 700, 850, 1008, 1195, 1435, 1693, 2004, 2390, 2808, 3303, 3910, 4584, 5372, 6328, 7387, 8619, 10106, 11757, 13675, 15961, 18508, 21464, 24948, 28845, 33345

Offset: 0

Ilya Gutkovskiy, May 17 2019

nonn

Weigh transform of A008480.

Cf. A000040, A008480, A129373, A308271.

g:= proc(n) option remember; (l-> add(i, i=l)!/
      mul(i!, i=l))(map(i-> i[2], ifactors(n)[2]))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2019

terms = 53; A[] = 1; Do[A[x] = (1 + x) Product[A[x^Prime[k]], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} (1 + x^k)^A008480(k).

A308284 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^3)A(x^5)A(x^7)* ... A(x^(2k-1))* ...

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 3, 3, 2, 4, 4, 4, 7, 8, 7, 9, 10, 11, 15, 15, 16, 23, 26, 24, 32, 36, 37, 47, 51, 54, 66, 71, 79, 99, 105, 108, 132, 148, 156, 184, 203, 219, 262, 282, 297, 358, 390, 417, 484, 531, 569, 654, 718, 773, 888, 962, 1037, 1198, 1303, 1390, 1592, 1740, 1868

Offset: 0

Ilya Gutkovskiy, May 18 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A074206, A129373, A308272, A308283.

terms = 65; A[] = 1; Do[A[x] = (1 + x) Product[A[x^(2 k - 1)], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} (1 + x^(2*k-1))^A074206(2*k-1).

A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

Original entry on oeis.org

1, -1, -1, 0, -1, 2, -1, 3, -1, 1, 4, -2, -4, -1, 7, -11, 0, -6, 3, -6, 9, 2, -6, 2, 5, 12, -16, 48, 4, -1, -28, 26, -30, 30, -62, 12, 16, -55, -9, -39, 18, -47, 103, -149, 87, -182, 210, -9, 45, 75, 225, -174, 39, 273, 11, 164, -224, 77, -105, 117, -703, 715, -678

Offset: 0

Seiichi Manyama, Nov 04 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A129374.

Cf. A074206, A129373, A321326.

G.f.: Product_{k>=1} (1 - x^k)^A074206(k) where A074206(n) is the number of ordered factorizations of n.

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, -2, 2, 1, 2, -3, 1, -3, -1, -7, 8, 4, 9, -7, 7, -7, 0, -21, 15, 2, 18, -23, 8, -25, -1, -43, 46, 17, 58, -34, 40, -41, 9, -98, 79, 10, 100, -98, 40, -123, -2, -191, 176, 43, 237, -136, 144, -192, 30, -362, 277, 12, 373, -314, 131, -457, -9, -606

Offset: 0

Seiichi Manyama, Nov 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A067856, A073707, A129373, A321326.

G.f.: Product_{k>0} (1 + x^k)^A067856(k).

Product_{k>0} A(x^k) = Product_{k>=0} (1 + x^(2^k))^(2^k). (Cf. A073707.)

cp's OEIS Frontend

A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

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A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

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A307605 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*... .

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A308272 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)*A(x^3)*A(x^5)* ... *A(x^prime(k))* ...

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A308284 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^3)*A(x^5)*A(x^7)* ... *A(x^(2*k-1))* ...

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A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A129375 E.g.f. satisfies: A(x) = exp(x) * A(x^2)A(x^3)A(x^4)...A(x^n)*...

A307605 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

A318767 G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

A308272 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)A(x^3)A(x^5)* ... A(x^prime(k)) ...

A308284 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^3)A(x^5)A(x^7)* ... A(x^(2k-1))* ...

A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).