cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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* ...

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

Views

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

Comments

Weigh transform of A050369.

Examples

			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 + ...

Links

Crossrefs

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

Programs

  • Mathematica
    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]

Formula

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

A307606 G.f. A(x) satisfies: A(x) = ((1 + x)/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, 2, 6, 16, 46, 104, 268, 596, 1406, 3060, 6812, 14356, 30948, 63660, 132328, 267164, 541678, 1072000, 2127052, 4140340, 8060588, 15458948, 29602504, 55990780, 105693252, 197422424, 367793952, 679206200, 1250557768, 2284986580, 4162202864, 7530956532, 13583095710
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

Comments

Convolution of A307604 and A307605.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 46*x^4 + 104*x^5 + 268*x^6 + 596*x^7 + 1406*x^8 + 3060*x^9 + ...

Links

Crossrefs

Cf. A050369, A074206, A307604, A307605, A318767.

Programs

  • Mathematica
    terms = 32; A[] = 1; Do[A[x] = (1 + x)/(1 - x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*A074206(k)).
a(n) ~ ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(50*(2+r))) * exp(12/625 + 2^(1/(2+r) - 1) * (2+r) * ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(2+r)) / (zeta'(r)^(1/(2+r)) * (1+r)) * n^((1+r)/(2+r))) / (A^(144/625) * 2^((3 + 2*r)/(50*(2 + r))) * zeta'(r)^(1/(50*(2+r))) * sqrt(Pi*(2+r)) * n^(1/2 + 1/(50*(2+r)))), where r = A107311 is the root of the equation zeta(r)=2 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 18 2021

A307607 a(n) = 1 + Sum_{d|n, d > 1} d^2*a(n/d).

Original entry on oeis.org

1, 5, 10, 37, 26, 122, 50, 293, 172, 330, 122, 1306, 170, 642, 710, 2341, 290, 2876, 362, 3562, 1382, 1578, 530, 13082, 1276, 2202, 3088, 6946, 842, 12822, 962, 18725, 3398, 3762, 3750, 37756, 1370, 4698, 4742, 35818, 1682, 25014, 1850, 17098, 17072, 6882
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

Links

Crossrefs

Cf. A074206, A197953, A307604.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + DivisorSum[n, #^2 a[n/#] &, # > 1 &]; Table[a[n], {n, 1, 46}]
  • PARI
    a(n) = 1 + sumdiv(n, d, if (d>1, d^2*a(n/d))); \\ Michel Marcus, Apr 20 2019

Formula

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k*A074206(k))) = Sum_{n>=1} a(n)*x^n/n.

A307615 E.g.f. A(x) satisfies: A(x) = exp(x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, 1, 5, 31, 337, 2741, 40621, 474475, 8461601, 132034537, 2648537461, 50079699671, 1204884343345, 26450428964701, 697107087763997, 17873985363570211, 526080367468142401, 15060611189639187665, 487251625325328212581, 15494976568071805188367, 545902629556769672596241
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 337*x^4/4! + 2741*x^5/5! + 40621*x^6/6! + 474475*x^7/7! + 8461601*x^8/8! + 132034537*x^9/9! + ...

Links

Crossrefs

Cf. A050369, A074206, A129375, A307604.

Programs

  • Mathematica
    terms = 20; A[] = 1; Do[A[x] = Exp[x] Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f.: exp(Sum_{k>=1} A050369(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} A074206(k)*k*k!*binomial(n-1,k-1)*a(n-k).
a(n) ~ (-Gamma(2+r)/zeta'(r))^(1/(4 + 2*r)) * exp(-n + 12/25 + n^(1 - 1/(2+r)) * (2+r) * (-Gamma(2+r)/zeta'(r))^(1/(2+r)) / (1+r)) * n^(n - 1/(4 + 2*r)) / sqrt(2+r), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Aug 09 2021

A309174 E.g.f. A(x) satisfies: A(x) = (1/(1 - x)) * Product_{k>=2} A(x^k)^(1/k).

Original entry on oeis.org

1, 1, 3, 11, 65, 369, 3139, 24667, 268449, 2777345, 34932131, 432114891, 6790407073, 97969389361, 1671204338595, 28382893729499, 557174580764609, 10512263160373377, 228918738980395459, 4817409763554888715, 115117419384636141441, 2688602544800222293361
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2019

Keywords

Crossrefs

Cf. A074206, A129374, A307604, A309175.

Programs

  • Mathematica
    terms = 21; A[] = 1; Do[A[x] = 1/(1 - x) Product[A[x^k]^(1/k), {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A074206(k)/k).
Showing 1-5 of 5 results.

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