A145268 -id:A145268 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Apr 02 2018

nonn

			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 * ...).

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

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

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 *)

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

A192478 G.f. satisfies: A(x) = xProduct_{n>=1} 1/(1 - xA(x)^n).

Original entry on oeis.org

1, 0, 1, 1, 3, 5, 14, 28, 75, 170, 443, 1076, 2795, 7046, 18398, 47458, 124793, 326905, 865853, 2293103, 6114688, 16327765, 43800590, 117720693, 317452606, 857742087, 2323588888, 6306229600, 17151172495, 46725729232, 127522337636, 348562660876

Offset: 1

Paul D. Hanna, Jul 01 2011

nonn

Related q-series identity (Euler):

Product_{n>=1} 1/(1-x*q^n) = Sum_{n>=0} x^n*q^n / Product_{k=1..n} (1-q^k); here q=A(x).

			G.f.: A(x) = x + x^3 + x^4 + 3*x^5 + 5*x^6 + 14*x^7 + 28*x^8 + 75*x^9 +...
The g.f. A = A(x) satisfies the relations:
A = x/((1 - x*A)*(1 - x*A^2)*(1 - x*A^3)*(1 - x*A^4)*...),
A = x*(1 + x*A/(1-A) + x^2*A^2/((1-A)*(1-A^2)) + x^3*A^3/((1-A)*(1-A^2)*(1-A^3)) +...).
A = x*(1 + x*A/((1-A)*(1-x*A)) + x^2*A^4/((1-A)*(1-x*A)*(1-A^2)*(1-x*A^2)) + x^3*A^9/((1-A)*(1-x*A)*(1-A^2)*(1-x*A^2)*(1-A^3)*(1-x*A^3)) +...).

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

Cf. A145268, A192477.

nmax = 40; A[] = 0; Do[A[x] = x/Product[1 - x*A[x]^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; Rest[CoefficientList[A[x], x]] (* Vaclav Kotesovec, Sep 29 2023 *)
(* Calculation of constants {d,c}: *) {1/r, s*Sqrt[r*((1 - r)*Log[1 - s] + (1 - 2*r)*Log[s] + (1 - r)* QPolyGamma[0, Log[r]/Log[s], s]) / (2*Pi*Log[s]*(2*(1 - r)*r - s^3*Derivative[0, 2][QPochhammer][r, s]))]} /. FindRoot[{(1 - r)*r == s*QPochhammer[r, s], s^2*Derivative[0, 1][QPochhammer][r, s] == (r - 1)*r}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 29 2023 *)

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

{a(n) = my(A=x+x^2); for(i=1, n, A = x*sum(m=0, n, x^m*A^m/prod(k=1, m, (1-A^k +x*O(x^n) )))); polcoeff(A, n)}

{a(n) = my(A=x+x^2); for(i=1, n, A = x*sum(m=0, sqrtint(n+1), A^(m^2)*x^m/prod(k=1, m, (1-A^k)*(1-x*A^k +x*O(x^n) )))); polcoeff(A, n)}

G.f. satisfies: A(x) = x*Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1 - A(x)^k).
G.f. satisfies: A(x) = x*Sum_{n>=0} x^n*A(x)^(n^2) / (Product_{k=1..n} (1 - A(x)^k)*(1 - x*A(x)^k)) due to Cauchy's identity.
a(n) ~ c * d^n / n^(3/2), where d = 2.8676696539306775593459072106295374201870092501605360805431348994... and c = 0.1427560675782905659177127310754143738629932908335931082457954696... - Vaclav Kotesovec, Sep 29 2023

A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^nA(x)^n x^(n^2) / Product_{k=1..n} (1 - 3x^k)(1 - x^k*A(x)).

Original entry on oeis.org

1, 3, 21, 147, 1074, 8076, 62454, 494292, 3990378, 32756142, 272715870, 2297982828, 19563641319, 168036314862, 1454458825605, 12674387617266, 111104771086812, 979101922849230, 8668964794053837, 77080072176742422, 687976906966730076, 6161811541538326680

Offset: 0

Paul D. Hanna, Feb 10 2012

nonn

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 147*x^3 + 1074*x^4 + 8076*x^5 +...
where the g.f. satisfies:
(0) A(x) = 1 + 3*x*A(x)/((1-3*x)*(1-x*A(x))) + 9*x^4*A(x)^2/((1-3*x)*(1-3*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 27*x^9*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
(1) A(x) = 1 + 3*x*A(x)/(1-3*x) + 3*x^2*A(x)^2/((1-3*x)*(1-3*x^2)) + 3*x^3*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)) +...
(2) A(x) = 1 + 3*x*A(x)/(1-x*A(x)) + 9*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 27*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Cf. A145268, A196150, A206637.

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*3^m*A^m/prod(k=1, m, (1-3*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3*x^m*A^m/prod(k=1, m, (1-3*x^k+x*O(x^n))))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3^m*x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0,35,print1(a(n),", "))

G.f. satisfies the identities:
(1) A(x) = 1 + Sum_{n>=1} 3*x^n*A(x)^n / Product_{k=1..n} (1 - 3*x^k).
(2) A(x) = 1 + Sum_{n>=1} 3^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)).

A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2n) / Product_{k=1..n} (1 - x^kA(x))^2.

Original entry on oeis.org

1, 1, 4, 18, 91, 489, 2751, 15985, 95218, 578324, 3568084, 22299964, 140885754, 898292262, 5772951668, 37355908797, 243184468271, 1591567315702, 10465836784159, 69114490893596, 458171948148640, 3047865264442504, 20339282134624054, 136122586785459512

Offset: 0

Paul D. Hanna, Feb 11 2012

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 91*x^4 + 489*x^5 + 2751*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + x*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^4/((1-x*A(x))^2*(1-x^2*A(x))^2) + x^9*A(x)^6/((1-x*A(x))^2*(1-x^2*A(x))^2*(1-x^3*A(x))^2) +...
(1) A(x) = 1 + x*A(x)^2/(1-x*A(x)) + x^2*A(x)^3/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)^4/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Cf. A196150, A145268, A206637.

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

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(m+1)/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

G.f. satisfies the identities:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n+1) / Product_{k=1..n} (1 - x^k*A(x)).
(2) A(x) = 1/(1 - Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1 - x^k*A(x)) ).

A302288 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)).

Original entry on oeis.org

1, 1, 4, 14, 55, 217, 908, 3864, 16894, 75078, 338862, 1548055, 7147427, 33294790, 156305144, 738753341, 3512431392, 16788169689, 80619590577, 388785776751, 1882063496033, 9142361671588, 44550166132194, 217716111661799, 1066792279046783, 5239947708977474, 25795965431819883

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 55*x^4 + 217*x^5 + 908*x^6 + 3864*x^7 + 16894*x^8 + 75078*x^9 + 338862*x^10 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - 2*x^2*A(x)) * (1 - 3*x^3*A(x)) * (1 - 4*x^4*A(x)) * ...).

Cf. A006906, A145268, A298261, A301577, A302171, A302289.

A262784 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x)^n * (x^n/n) / (1 + x^n) ).

Original entry on oeis.org

1, 1, 1, 3, 7, 18, 51, 147, 431, 1295, 3954, 12219, 38174, 120373, 382559, 1224173, 3940964, 12754732, 41476011, 135446194, 444016332, 1460619970, 4819995330, 15951789030, 52932499724, 176073789065, 587010353666, 1961118581900, 6564548926877, 22013613082514, 73945428978360, 248781470115309

Offset: 0

Paul D. Hanna, Oct 01 2015

nonn

Compare to: exp( Sum_{n>=1} (x^n/n)/(1+x^n) ) = Sum_{n>=0} x^(n*(n+1)/2).

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 18*x^5 + 51*x^6 + 147*x^7 +...
where
log(A(x)) = A(x)*x/(1+x) + A(x)^2*(x^2/2)/(1+x^2) + A(x)^3*(x^3/3)/(1+x^3) + A(x)^4*(x^4/4)/(1+x^4) + A(x)^5*(x^5/5)/(1+x^5) +...
explicitly,
log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 56*x^5/5 + 187*x^6/6 + 617*x^7/7 + 2033*x^8/8 + 6811*x^9/9 + 22906*x^10/10 +...

Cf. A145268.

{a(n) = local(A=1+x); for(i=1,n, A = exp( sum(k=1,n, A^k*x^k/k/(1+x^k +x*O(x^n)))));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

cp's OEIS Frontend

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

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A192478 G.f. satisfies: A(x) = x*Product_{n>=1} 1/(1 - x*A(x)^n).

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A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)).

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A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2*n) / Product_{k=1..n} (1 - x^k*A(x))^2.

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A302288 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - k*x^k*A(x)).

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A262784 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x)^n * (x^n/n) / (1 + x^n) ).

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A192478 G.f. satisfies: A(x) = xProduct_{n>=1} 1/(1 - xA(x)^n).

A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^nA(x)^n x^(n^2) / Product_{k=1..n} (1 - 3x^k)(1 - x^k*A(x)).

A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2n) / Product_{k=1..n} (1 - x^kA(x))^2.

A302288 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)).