A200537
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,k) is the coefficient of x^k in (1+3*x+2*x^2)^n.
Original entry on oeis.org
1, 1, 9, 13, 40, 72, 144, 252, 432, 720, 1152, 1872, 2880, 4608, 6912, 10944, 16128, 25344, 36864, 57600, 82944, 129024, 184320, 285696, 405504, 626688, 884736, 1363968, 1916928, 2949120, 4128768, 6340608, 8847360, 13565952, 18874368, 28901376, 40108032, 61341696, 84934656, 129761280
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 13*x^3 + 40*x^4 + 72*x^5 + 144*x^6 +...
The logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + 3^2*x + 2^2*x^2)/A(x) * x +
(1 + 6^2*x + 13^2*x^2 + 12^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 +
(1 + 9^2*x + 62^2*x^2 + 63^2*x^3 + 66^2*x^4 + 36^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 +
(1 + 12^2*x + 62^2*x^2 + 180^2*x^3 + 321^2*x^4 + 360^2*x^5 + 248^2*x^6 + 96^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 +
(1 + 15^2*x + 100^2*x^2 + 390^2*x^3 + 985^2*x^4 + 1683^2*x^5 + 1970^2*x^6 + 1560^2*x^7 + 800^2*x^8 + 240^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients A200536(n,k) in (1+3*x+2*x^2)^n.
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CoefficientList[Series[(1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / (1-2*x^2)^2, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 11 2015 *)
Flatten[{1,1,9,13,40,Table[FullSimplify[9 * 2^(n/2-4) * (-8 - 7*Sqrt[2] + 4*n + 3*Sqrt[2]*n + (-1)^(n+1)*(8 - 7*Sqrt[2] + (-4 + 3*Sqrt[2])*n))],{n,5,40}]}] (* Vaclav Kotesovec, Feb 11 2015 *)
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{a(n)=polcoeff((1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / ((1-2*x^2)^2+x*O(x^n)),n)}
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, n, polcoeff((1+3*x+2*x^2+x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n)));polcoeff(A, n)}
A251689
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,2*n-k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,2*n-k) is the coefficient of x^k in (2+3*x+x^2)^n.
Original entry on oeis.org
1, 4, 9, 37, 40, 153, 144, 468, 432, 1260, 1152, 3168, 2880, 7632, 6912, 17856, 16128, 40896, 36864, 92160, 82944, 205056, 184320, 451584, 405504, 986112, 884736, 2138112, 1916928, 4608000, 4128768, 9879552, 8847360, 21086208, 18874368, 44826624, 40108032, 94961664, 84934656, 200540160
Offset: 0
G.f.: A(x) = 1 + 4*x + 9*x^2 + 37*x^3 + 40*x^4 + 153*x^5 + 144*x^6 +...
The logarithm of the g.f. A(x) equals the series:
log(A(x)) = (2^2 + 3^2*x + x^2)/A(x) * x +
(4^2 + 12^2*x + 13^2*x^2 + 6^2*x^3 + x^4)/A(x)^2 * x^2/2 +
(8^2 + 36^2*x + 66^2*x^2 + 63^2*x^3 + 33^2*x^4 + 9^2*x^5 + x^6)/A(x)^3 * x^3/3 +
(16^2 + 96^2*x + 248^2*x^2 + 360^2*x^3 + 321^2*x^4 + 180^2*x^5 + 62^2*x^6 + 12^2*x^7 + x^8)/A(x)^4 * x^4/4 +
(32^2 + 240^2*x + 800^2*x^2 + 1560^2*x^3 + 1970^2*x^4 + 1683^2*x^5 + 985^2*x^6 + 390^2*x^7 + 100^2*x^8 + 15^2*x^9 + x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients A200536(n,2*n-k) in (2+3*x+x^2)^n.
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{a(n)=polcoeff( (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / ((1-2*x^2)^2 +x*O(x^n)), n)}
for(n=0,40,print1(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, n, polcoeff((2+3*x+x^2+x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n))); polcoeff(A, n)}
for(n=0,40,print1(a(n),", "))
A218619
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..2*n} A200536(n,k)^2 * x^k * A(x)^(2*k) ), where A200536(n,k) = [x^k] (1 + 3*x + 2*x^2)^n.
Original entry on oeis.org
1, 1, 11, 72, 734, 6994, 74641, 803196, 8989482, 102192197, 1184211027, 13897707080, 165052834584, 1978844990494, 23924151189858, 291313067897212, 3569576082827250, 43981925261314302, 544590342185545146, 6772925262506494672, 84567358373934285042
Offset: 0
G.f.: A(x) = 1 + x + 11*x^2 + 72*x^3 + 734*x^4 + 6994*x^5 + 74641*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + 3^2*x*A^2 + 2^2*x^2*A^4)*x*A +
(1 + 6^2*x*A^2 + 13^2*x^2*A^4 + 12^2*x^3*A^6 + 4^2*x^4*A^8)*x^2*A^2/2 +
(1 + 9^2*x*A^2 + 33^2*x^2*A^4 + 63^2*x^3*A^6 + 66^2*x^4*A^8 + 36^2*x^5*A^10 + 8^2*x^6*A^12)*x^3*A^3/3 +
(1 + 12^2*x*A^2 + 62^2*x^2*A^4 + 180^2*x^3*A^6 + 321^2*x^4*A^8 + 360^2*x^5*A^10 + 248^2*x^6*A^12 + 96^2*x^7*A^14 + 16^2*x^8*A^16)*x^4*A^4/4 +...
which involves the squares of the trinomial coefficients A200536(n,k):
1;
1, 3, 2;
1, 6, 13, 12, 4;
1, 9, 33, 63, 66, 36, 8;
1, 12, 62, 180, 321, 360, 248, 96, 16;
1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32; ...
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/* G.f. A(x) using the squares of the trinomial coefficients */
{A200536(n, k)=polcoeff((1 + 3*x + 2*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, A200536(m, k)^2*x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
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{a(n)=local(A=sqrt(serreverse( x*(1-2*x^2)^4/((1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3+x*O(x^n)))^2 )/x));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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