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 1-4 of 4 results.

A066398 Reversion of g.f. (with constant term included) for partition numbers.

Original entry on oeis.org

1, -1, 0, 2, -3, 0, 5, 0, -21, 14, 117, -342, 210, 935, -2565, 1864, 2751, -3945, -8074, 4046, 108927, -333832, 246895, 887040, -2764795, 3062749, -1372098, 4775900, -9367698, -55130625, 299939766, -537241936, -140898285, 2464380030, -4060507784, 193070394
Offset: 0

Views

Author

N. J. A. Sloane, Dec 25 2001

Keywords

Comments

See A301624 for the corresponding series reversion for the plane partition numbers A000219. - Peter Bala, Feb 09 2020

Links

Crossrefs

Cf. A000041, A000203, A007312, A301624.

Programs

  • Maple
    with(numtheory):
Order := 36:
Gser := solve(series(x*exp(add(sigma[1](n)*x^n/n, n = 1..35)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..35); # Peter Bala, Feb 09 2020
  • Mathematica
    nmax = 34; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Product[ 1 - x^k*A[x]^k, {k, 1, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

Formula

The o.g.f. A(x) = 1 - x + 2*x^3 - 3*x^4 + 5*x^6 - ... satisfies [x^n](1/A(x))^n = sigma(n) = A000203(n) for n >= 1. - Peter Bala, Aug 23 2015
G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k). - Ilya Gutkovskiy, Mar 21 2018

A302171 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x))^k.

Original entry on oeis.org

1, 1, 4, 14, 54, 213, 880, 3724, 16143, 71227, 319067, 1447160, 6633530, 30682425, 143028870, 671293632, 3169572659, 15044993968, 71752624923, 343658572717, 1652266087698, 7971518032791, 38581202763318, 187269381724629, 911404238805468, 4446493502832481, 21742327471261176
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 02 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 213*x^5 + 880*x^6 + 3724*x^7 + 16143*x^8 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x))^2 * (1 - x^3*A(x))^3 * ...).

Links

Crossrefs

Cf. A000219, A145267, A145268, A190862, A298260, A298261, A301455, A301624, A301831.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1/Product[(1 - x^k*A[x])^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 5.177446537296361283814259811908762546749... and c = 0.81395777803098291048009263980507199... - Vaclav Kotesovec, Sep 27 2023
Radius of convergence r = 0.1931454033945844258723936803941781838... = 1/d and A(r) = 2.2252305561396523944672847657756264073... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 - r^n*A(r)) and (2) A(r) = 1 / Product_{n>=1} (1 - r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^k*A(x)^k)/(1 - x^k*A(x)^k))^k.

Original entry on oeis.org

1, 2, 10, 60, 398, 2820, 20892, 159868, 1253758, 10024070, 81400672, 669532924, 5566386324, 46701736772, 394910202608, 3362210548344, 28797181196766, 247955463799812, 2145088563952510, 18636002388075260, 162523319555310664, 1422259430668179592, 12485554521209720492, 109922263517662775292
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2018

Keywords

Examples

			G.f. A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 398*x^4 + 2820*x^5 + 20892*x^6 + 159868*x^7 + 1253758*x^8 + ...
G.f. A(x) satisfies: A(x) = ((1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * ...)/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = 2*x + 16*x^2/2 + 128*x^3/3 + 1056*x^4/4 + 8952*x^5/5 + 77200*x^6/6 + 673948*x^7/7 + 5937792*x^8/8 + ... + A270924(n)*x^n/n + ...

Crossrefs

Cf. A006195, A066398, A109085, A156616, A181315, A270924, A278428, A301455, A301456, A301624.

A301831 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, 0, 0, 6, -16, 16, -34, 217, -681, 1343, -3466, 13370, -42380, 109477, -312448, 1040248, -3267138, 9447529, -28367596, 90504001, -283611105, 861087913, -2654231074, 8386506600, -26359974392, 81902319183, -256179313766, 809890745232, -2557697524240, 8046530976599
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2018

Keywords

Examples

			G.f. A(x) = 1 - x + 6*x^4 - 16*x^5 + 16*x^6 - 34*x^7 + 217*x^8 - 681*x^9 + 1343*x^10 - 3466*x^11 + ...
log(A(x)) = -x - x^2/2 - x^3/3 + 23*x^4/4 - 51*x^5/5 + 35*x^6/6 - 197*x^7/7 + ... + A281266(n)*x^n/n + ...

Crossrefs

Cf. A109085, A181315, A255528, A281266, A301455, A301456, A301624.

Formula

G.f. satisfies: A(x) = exp(Sum_{k>=1} (-1)^k*x^k*A(x)^k/(k*(1 - x^k*A(x)^k)^2)).
a(n) = [x^n] (Sum_{k>=0} A255528(k)*x^k)^(n+1)/(n + 1).
Showing 1-4 of 4 results.

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