A199257
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 * x^k * A(x)^k]* x^n/n ).
Original entry on oeis.org
1, 1, 5, 18, 86, 408, 2075, 10787, 57655, 313643, 1733450, 9700574, 54867895, 313145033, 1801150861, 10430094658, 60758092753, 355795743385, 2093295146379, 12367548160650, 73346850194969, 436486017193373, 2605656191324094, 15599323024019360, 93634195155551584
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 86*x^4 + 408*x^5 + 2075*x^6 +...
such that A(x) = G(x*A(x)) where G(x) = (1-x+x^2)*(1+x^2)^2/(1-x)^2:
G(x) = 1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 12*x^5 + 16*x^6 + 20*x^7 + 24*x^8 +...
...
Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + 2^2*A + A^2)*x +
(1 + 4^2*A + 6^2*A^2 + 4^2*A^3 + A^4)*x^2/2 +
(1 + 6^2*A + 15^2*A^2 + 20^2*A^3 + 15^2*A^4 + 6^2*A^5 + A^6)*x^3/3 +
(1 + 8^2*A + 28^2*A^2 + 56^2*A^3 + 70^2*A^4 + 56^2*A^5 + 28^2*A^6 + 8^2*A^7 + A^8)*x^4/4 +
(1 + 10^2*A + 45^2*A^2 + 120^2*A^3 + 210^2*A^4 + 252^2*A^5 + 210^2*A^6 + 120^2*A^7 + 45^2*A^8 + 10^2*A^9 + A^10)*x^5/5 +...
which involves the squares of binomial coefficients C(2*n,k).
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{a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)^2/((1-x+x^2)*(1+x^2)^2+x*O(x^n))); polcoeff(A, n)}
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, binomial(2*m, k)^2 *x^k*A^k) *x^m/m)+x*O(x^n))); polcoeff(A, n)}
A200475
G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^(2*k)) * x^n*A(x)^n/n ), where A027907 is the triangle of trinomial coefficients.
Original entry on oeis.org
1, 1, 3, 13, 65, 350, 1981, 11627, 70132, 432090, 2707595, 17202779, 110563543, 717547090, 4695774335, 30952628861, 205318395288, 1369539030021, 9180527051187, 61813112864984, 417850301293691, 2834802846097200, 19294989810689802, 131723105933867817, 901709774424393614
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 65*x^4 + 350*x^5 + 1981*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A^2 + x^2*A^4)*x*A +
(1 + 2^2*x*A^2 + 3^2*x^2*A^4 + 2^2*x^3*A^6 + x^4*A^8)*x^2*A^2/2 +
(1 + 3^2*x*A^2 + 6^2*x^2*A^4 + 7^2*x^3*A^6 + 6^2*x^4*A^8 + 3^2*x^5*A^10 + x^6*A^12)*x^3*A^3/3 +
(1 + 4^2*x*A^2 + 10^2*x^2*A^4 + 16^2*x^3*A^6 + 19^2*x^4*A^8 + 16^2*x^5*A^10 + 10^2*x^6*A^12 + 4^2*x^7*A^14 + x^8*A^16)*x^4*A^4/4 +
(1 + 5^2*x*A^2 + 15^2*x^2*A^4 + 30^2*x^3*A^6 + 45^2*x^4*A^8 + 51^2*x^5*A^10 + 45^2*x^6*A^12 + 30^2*x^7*A^14 + 15^2*x^8*A^16 + 5^2*x^9*A^18 + x^10*A^20)*x^5*A^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).
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{a(n)=local(A=1+x);for(i=1,n,A=(1-x*A^2+x^3*A^6-x^5*A^10+x^6*A^12)/(1-x*A^2+x*O(x^n))^2);polcoeff(A,n)}
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/* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}
A200377
G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k / A(x)^k) * x^n/n ).
Original entry on oeis.org
1, 1, 2, 4, 7, 11, 19, 34, 61, 106, 181, 311, 543, 955, 1668, 2885, 4980, 8650, 15114, 26391, 45845, 79385, 137718, 239866, 418338, 727926, 1263097, 2191463, 3810775, 6638258, 11556361, 20078960, 34855400, 60567092, 105405431, 183483906, 319039355, 554158992, 962743619, 1674359119, 2913758685, 5068194691
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 19*x^6 + 34*x^7 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x/A + x^2/A^2)*x +
(1 + 2^2*x/A + 3^2*x^2/A^2 + 2^2*x^3/A^3 + x^4/A^4)*x^2/2 +
(1 + 3^2*x/A + 6^2*x^2/A^2 + 7^2*x^3/A^3 + 6^2*x^4/A^4 + 3^2*x^5/A^5 + x^6/A^6)*x^3/3 +
(1 + 4^2*x/A + 10^2*x^2/A^2 + 16^2*x^3/A^3 + 19^2*x^4/A^4 + 16^2*x^5/A^5 + 10^2*x^6/A^6 + 4^2*x^7/A^7 + x^8/A^8)*x^4/4 +
(1 + 5^2*x/A + 15^2*x^2/A^2 + 30^2*x^3/A^3 + 45^2*x^4/A^4 + 51^2*x^5/A^5 + 45^2*x^6/A^6 + 30^2*x^7/A^7 + 15^2*x^8/A^8 + 5^2*x^9/A^9 + x^10/A^10)*x^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).
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/* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k/(A+x*O(x^n))^k) *x^m/m))); polcoeff(A, n)}
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