A251689 -id:A251689 - cp's OEIS frontend

A200537 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} A200536(n,k)^2 x^k] / A(x)^n * x^n/n ), where A200536(n,k) is the coefficient of x^k in (1+3x+2x^2)^n.

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

Paul D. Hanna, Nov 18 2011

nonn

			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.

Cf. A200536, A200535, A199257, A251689.

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 *)

{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)}

{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)}

G.f.: (1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / (1-2*x^2)^2.

A251688 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2x)^n(1+3*x)^n.

Original entry on oeis.org

1, 1, 25, 61, 336, 1200, 3600, 13500, 32400, 118800, 259200, 939600, 1944000, 6998400, 13996800, 50155200, 97977600, 349920000, 671846400, 2393452800, 4534963200, 16124313600, 30233088000, 107327462400, 199538380800, 707454259200, 1306069401600, 4625662464000, 8489451110400

Offset: 0

Paul D. Hanna, Feb 25 2015

nonn

More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + a*x)^n*(1 + b*x)^n, then G(x) = (1+x)*(1 + a^2*x^2)*(1 + b^2*x^2)*(1 + a^2*b^2*x^3) / (1 - a*b*x^2)^2; here a=2, b=3.

More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (p + q*x)^n*(r + s*x)^n, then G(x) = (1 + p^2*r^2*x)*(1 + p^2*s^2*x^2)*(1 + q^2*r^2*x^2)*(1 + q^2*s^2*x^3) / (1 - p*q*r*s*x^2)^2.

			G.f.: A(x) = 1 + x + 25*x^2 + 61*x^3 + 336*x^4 + 1200*x^5 + 3600*x^6 +...
where
log(A(x)) = (1 + 5^2*x + 6^2*x^2)/A(x) * x +
(1 + 10^2*x + 37^2*x^2 + 60^2*x^3 + 36^2*x^4)/A(x)^2 * x^2/2 +
(1 + 15^2*x + 93^2*x^2 + 305^2*x^3 + 558^2*x^4 + 540^2*x^5 + 216^2*x^6)/A(x)^3 * x^3/3 +
(1 + 20^2*x + 174^2*x^2 + 860^2*x^3 + 2641^2*x^4 + 5160^2*x^5 + 6264^2*x^6 + 4320^2*x^7 + 1296^2*x^8)/A(x)^4 * x^4/4 +
(1 + 25^2*x + 280^2*x^2 + 1850^2*x^3 + 7985^2*x^4 + 23525^2*x^5 + 47910^2*x^6 + 66600^2*x^7 + 60480^2*x^8 + 32400^2*x^9 + 7776^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients in (1 + 5*x + 6*x^2)^n.

Cf. A251689, A200537, A251687.

{a(n)=polcoeff( (1+x)*(1+4*x^2)*(1+9*x^2)*(1+36*x^3) / ((1-6*x^2)^2 +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff(((1+2*x)*(1+3*x) +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), ", "))

G.f.: (1+x)*(1+4*x^2)*(1+9*x^2)*(1+36*x^3) / (1-6*x^2)^2.

A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.

Original entry on oeis.org

1, 1, 1, 5, 8, 8, 16, 28, 48, 80, 128, 208, 320, 512, 768, 1216, 1792, 2816, 4096, 6400, 9216, 14336, 20480, 31744, 45056, 69632, 98304, 151552, 212992, 327680, 458752, 704512, 983040, 1507328, 2097152, 3211264, 4456448, 6815744, 9437184, 14417920, 19922944, 30408704, 41943040

Offset: 0

Paul D. Hanna, Mar 03 2015

nonn

More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (p + q*x + r*x^2)^n, then G(x) = (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / (1-p*r*x^2)^2.

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 8*x^4 + 8*x^5 + 16*x^6 + 28*x^7 +...
where
log(A(x)) = (1 + x + 2^2*x^2)/A(x) * x +
(1 + 2^2*x + 5^2*x^2 + 4^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 +
(1 + 3^2*x + 9^2*x^2 + 13^2*x^3 + 18^2*x^4 + 12^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 +
(1 + 4^2*x + 14^2*x^2 + 28^2*x^3 + 49^2*x^4 + 56^2*x^5 + 56^2*x^6 + 32^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 +
(1 + 5^2*x + 20^2*x^2 + 50^2*x^3 + 105^2*x^4 + 161^2*x^5 + 210^2*x^6 + 200^2*x^7 + 160^2*x^8 + 80^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients in (1 + x + 2*x^2)^n - see triangle A084600.

Cf. A084600, A251688, A251689, A200537.

/* By Definition: */
{a(n,p=1,q=1,r=2)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff((p + q*x + r*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, 50, print1(a(n), ", "))

/* By G.F. Identity (faster): */
{a(n,p=1,q=1,r=2)=polcoeff( (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / ((1-p*r*x^2)^2 +x*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))

G.f.: (1 + x)*(1 + 4*x^3)*(1 - 3*x^2 + 4*x^4) / (1 - 2*x^2)^2.

cp's OEIS Frontend

A200537 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} A200536(n,k)^2 x^k] / A(x)^n * x^n/n ), where A200536(n,k) is the coefficient of x^k in (1+3x+2x^2)^n.

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A251688 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2x)^n(1+3*x)^n.

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A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.

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cp's OEIS Frontend

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.

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A251688 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2*x)^n*(1+3*x)^n.

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A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.

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A200537 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} A200536(n,k)^2 x^k] / A(x)^n * x^n/n ), where A200536(n,k) is the coefficient of x^k in (1+3x+2x^2)^n.

A251688 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2x)^n(1+3*x)^n.

A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.