A301456 -id:A301456 - cp's OEIS frontend

A301455 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x)^k)^k.

1, 1, 4, 16, 74, 360, 1840, 9698, 52409, 288697, 1615275, 9153850, 52434770, 303104532, 1765920785, 10358843904, 61129390652, 362650003202, 2161590275029, 12938838382316, 77745063802045, 468760264760369, 2835272729215565, 17198394229862818, 104598950726341920, 637709136315071504

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 74*x^4 + 360*x^5 + 1840*x^6 + 9698*x^7 + 52409*x^8 + 288697*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 215*x^4/4 + 1251*x^5/5 + 7459*x^6/6 + 44885*x^7/7 + 272727*x^8/8 + ... + A255672(n)*x^n/n + ...

Cf. A000219, A001157, A109085, A145268, A255672, A301456.

G.f. A(x) satisfies: A(x) = exp(Sum_{k>=1} sigma_2(k)*x^k*A(x)^k/k).

A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

sign

			G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
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^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

From Peter Bala, Feb 09 2020: (Start)
A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).
Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)
A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219).  - Peter Bala, Feb 11 2020

A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)^k).

Original entry on oeis.org

1, 1, 3, 12, 48, 211, 970, 4594, 22311, 110473, 555561, 2829918, 14570666, 75708835, 396481070, 2090558864, 11089276706, 59135014252, 316836936662, 1704764660218, 9207671377450, 49904141524184, 271325301723223, 1479427708380368, 8088057338101442, 44325245804200151

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 48*x^4 + 211*x^5 + 970*x^6 + 4594*x^7 + 22311*x^8 + 110473*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + 2*x^2*A(x)^2) * (1 + 3*x^3*A(x)^3) * (1 + 4*x^4*A(x)^4) * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 137*x^4/4 + 726*x^5/5 + 3896*x^6/6 + 21071*x^7/7 + 115089*x^8/8 + ... + A297322(n)*x^n/n + ...

Cf. A022629, A181315, A297322, A301456, A301577.

A302287 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x))^k.

Original entry on oeis.org

1, 1, 3, 10, 31, 102, 342, 1167, 4046, 14213, 50464, 180847, 653296, 2376406, 8697194, 32002219, 118322499, 439364380, 1637827543, 6126870808, 22993190147, 86542625565, 326607659370, 1235650643059, 4685502714403, 17804713119018, 67790202024365, 258579199501709, 988012193672223

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 31*x^4 + 102*x^5 + 342*x^6 + 1167*x^7 + 4046*x^8 + 14213*x^9 + 50464*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2 * (1 + x^3*A(x))^3 * (1 + x^4*A(x))^4 * ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A026007, A145267, A298260, A301456, A302171.

nmax = 30; A[] = 0; Do[A[x] = 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 *)

a(n) ~ c * d^n / n^(3/2), where d = 4.01604513838270620496843653760987690323... and c = 2.07544072297996637757124624302382219... - Vaclav Kotesovec, Sep 27 2023

Radius of convergence r = 0.2490011853807768883971843288180859269 = 1/d and A(r) = 3.261386924996517219078267128734843819... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 + r^n*A(r)) and (2) A(r) = 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^kA(x)^k)/(1 - x^kA(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

Ilya Gutkovskiy, Mar 24 2018

nonn

			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 + ...

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

Ilya Gutkovskiy, Mar 27 2018

sign

			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 + ...

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

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).

A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)).

Original entry on oeis.org

1, 1, 3, 10, 30, 98, 323, 1083, 3684, 12710, 44272, 155608, 551259, 1965952, 7052990, 25436711, 92168542, 335376653, 1224991077, 4489818110, 16507728007, 60868469848, 225030777305, 833961333273, 3097594423724, 11529400593846, 42996077073284, 160632616725238, 601132116489719, 2253153800577748

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 30*x^4 + 98*x^5 + 323*x^6 + 1083*x^7 + 3684*x^8 + 12710*x^9 + 44272*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + 2*x^2*A(x)) * (1 + 3*x^3*A(x)) * (1 + 4*x^4*A(x)) * ...

Cf. A022629, A145267, A301456, A301577, A301578, A302287, A302288.

cp's OEIS Frontend

A301455 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x)^k)^k.

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A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

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Maple

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A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + k*x^k*A(x)^k).

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A302287 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x))^k.

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Mathematica

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

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

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A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + k*x^k*A(x)).

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A301578 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)^k).

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

A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)).