A216247
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^2*x^2)^n/n! * exp(-n*x*A(n^2*x^2)).
Original entry on oeis.org
1, 1, 1, 2, 15, 77, 354, 2509, 19369, 146840, 1321773, 13466872, 151177886, 1827144180, 23686530621, 335038346222, 4922031402993, 75768551557262, 1240672912146220, 21264349760865217, 384135346377854363, 7286741020161077248, 145190876484745421540
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 15*x^4 + 77*x^5 + 354*x^6 + 2509*x^7 +...
where
A(x) = 1 + x*A(x^2)*exp(-x*A(x^2)) + 2^2*x^2*A(2^2*x^2)^2/2!*exp(-2*x*A(2^2*x^2)) + 3^3*x^3*A(3^2*x^2)^3/3!*exp(-3*x*A(3^2*x^2)) + 4^4*x^4*A(4^2*x^2)^4/4!*exp(-4*x*A(4^2*x^2)) + 5^5*x^5*A(5^2*x^2)^5/5!*exp(-5*x*A(5^2*x^2)) +...
simplifies to a power series in x with integer coefficients.
-
{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A, x, k^2*x^2)^k/k!*exp(-k*x*subst(A, x, k^2*x^2)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A218671
O.g.f.: Sum_{n>=0} n^(2*n) * (1+n*x)^n * x^n/n! * exp(-n^2*x*(1+n*x)).
Original entry on oeis.org
1, 1, 8, 120, 2635, 76503, 2764957, 119634152, 6030195490, 347037131298, 22453144758980, 1613322276606404, 127466755375275614, 10983423290600347408, 1025046637630590359928, 103004615955568528609200, 11088429267977228122393005, 1273093489376335864500416685
Offset: 0
O.g.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 2635*x^4 + 76503*x^5 +...
where
A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^4*(1+2*x)^2*x^2/2!*exp(-2^2*x*(1+2*x)) + 3^6*(1+3*x)^3*x^3/3!*exp(-3^2*x*(1+3*x)) + 4^8*(1+4*x)^4*x^4/4!*exp(-4^2*x*(1+4*x)) + 5^10*(1+5*x)^5*x^5/5!*exp(-5^2*x*(1+5*x)) +...
simplifies to a power series in x with integer coefficients.
-
{a(n)= my(A=sum(k=0, n, k^(2*k)*(1+k*x)^k*x^k/k!*exp(-k^2*x*(1+k*x)+x*O(x^n)))); polcoef(A, n)}
for(n=0,30,print1(a(n),", "))
A219343
O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(n*x)).
Original entry on oeis.org
1, 1, 32, 3183, 650929, 226009218, 119298668857, 89086101638412, 89480710389500666, 116491795770107486363, 191172400354899371561288, 387419202671209086703674709, 956322827450633453264262285623, 2859815748552720894795327258080881, 10430012061189048036456303441601971435
Offset: 0
O.g.f.: A(x) = 1 + x + 32*x^2 + 3183*x^3 + 650929*x^4 + 226009218*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(2*x)^2/2!*exp(-2^3*x*A(2*x)) + 3^9*x^3*A(3*x)^3/3!*exp(-3^3*x*A(3*x)) + 4^12*x^4*A(4*x)^4/4!*exp(-4^3*x*A(4*x)) + 5^15*x^5*A(5*x)^5/5!*exp(-5^3*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
-
{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(3*k)*x^k*subst(A,x,k*x)^k/k!*exp(-k^3*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A223075
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A((2*n-1)*x)^n/n! * exp(-n*x*A((2*n-1)*x)).
Original entry on oeis.org
1, 1, 2, 13, 132, 2492, 76726, 4048401, 360486616, 54950141846, 14338767268684, 6424397920197266, 4947731418324541980, 6554636080888858780850, 14947781374271898418583534, 58699996835841575449007944393, 397110307362512858324163841229032
Offset: 0
O.g.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 132*x^4 + 2492*x^5 + 76726*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(3*x)^2/2!*exp(-2*x*A(3*x)) + 3^3*x^3*A(5*x)^3/3!*exp(-3*x*A(5*x)) + 4^4*x^4*A(7*x)^4/4!*exp(-4*x*A(7*x)) + 5^5*x^5*A(9*x)^5/5!*exp(-5*x*A(9*x)) +...
simplifies to a power series in x with integer coefficients.
-
{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A, x, (2*k-1)*x)^k/k!*exp(-k*x*subst(A, x, (2*k-1)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A223076
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(2*n*x)^n/n! * exp(-n*x*A(2*n*x)).
Original entry on oeis.org
1, 1, 3, 25, 433, 14929, 1009039, 134378493, 35413549073, 18529994604561, 19287258947192299, 39990414610486392193, 165330456559779835205073, 1363910437230335758822062353, 22464490025153709857947688719687, 739043653017364758151896078253911765
Offset: 0
O.g.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 433*x^4 + 14929*x^5 + 1009039*x^6 +...
where
A(x) = 1 + x*A(2*x)*exp(-x*A(2*x)) + 2^2*x^2*A(4*x)^2/2!*exp(-2*x*A(4*x)) + 3^3*x^3*A(6*x)^3/3!*exp(-3*x*A(6*x)) + 4^4*x^4*A(8*x)^4/4!*exp(-4*x*A(8*x)) + 5^5*x^5*A(10*x)^5/5!*exp(-5*x*A(10*x)) +...
simplifies to a power series in x with integer coefficients.
-
{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A, x, 2*k*x)^k/k!*exp(-k*x*subst(A, x, 2*k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
Comments