A198296
G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n) ).
Original entry on oeis.org
1, 1, 2, 3, 6, 8, 17, 22, 44, 62, 115, 154, 311, 409, 754, 1070, 1949, 2639, 4917, 6645, 12055, 16916, 29594, 40719, 73907, 100959, 176010, 248207, 429626, 594220, 1040624, 1436936, 2473555, 3486360, 5901887, 8233872, 14174779, 19689223, 33203829, 46967767
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...
such that, by definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)) + (x^3/3)/((1-x^3)*(1-3*x^3)) + (x^4/4)/((1-x^4)*(1-2*x^4)*(1-4*x^4)) + (x^5/5)/((1-x^5)*(1-5*x^5)) + (x^6/6)/((1-x^6)*(1-2*x^6)*(1-3*x^6)*(1-6*x^6)) +...+ (x^n/n)/Product_{d|n} (1-d*x^n) +...
Also, we have the identity:
log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2
+ (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3
+ (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4
+ (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5
+ (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...
+ exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...
Explicitly, the logarithm begins:
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 + 8*x^7/7 + 83*x^8/8 + 49*x^9/9 + 178*x^10/10 +...+ A198299(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*exp(sum(k=1,n\m,sigma(m,k)*x^(m*k)/k)+x*O(x^n)))),n)}
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*exp(sumdiv(m,d,-log(1-d*x^m+x*O(x^n)))))),n)}
A203318
G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) ) where Lucas(n) = A000032(n).
Original entry on oeis.org
1, 1, 2, 4, 9, 16, 36, 64, 135, 250, 504, 917, 1864, 3372, 6593, 12176, 23473, 42732, 82142, 149282, 283104, 516780, 967894, 1757865, 3291964, 5959633, 11039163, 20022457, 36908442, 66637739, 122512809, 220717328, 403499293, 726866565, 1322670966, 2376541137
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} F_n(x^n) * x^n/n )
where F_n(x) = exp( Sum_{k>=1} Lucas(n*k)*x^k/k ), which begin:
F_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 21*x^7 +...;
F_2(x) = 1 + 3*x + 8*x^2 + 21*x^3 + 55*x^4 + 144*x^5 + 377*x^6 +...;
F_3(x) = 1 + 4*x + 17*x^2 + 72*x^3 + 305*x^4 + 1292*x^5 + 5473*x^6 +...;
F_4(x) = 1 + 7*x + 48*x^2 + 329*x^3 + 2255*x^4 + 15456*x^5 +...;
F_5(x) = 1 + 11*x + 122*x^2 + 1353*x^3 + 15005*x^4 + 166408*x^5 +...;
F_6(x) = 1 + 18*x + 323*x^2 + 5796*x^3 + 104005*x^4 + 1866294*x^5 +...;
...
Also, F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where u = n-th root of unity:
F_1(x) = F(x) = 1/(1-x-x^2) = g.f. of the Fibonacci numbers;
F_2(x^2) = F(x)*F(-x) = 1/(1-3*x^2+x^4);
F_3(x^3) = F(x)*F(w*x)*F(w^2*x) = 1/(1-4*x^3-x^6) where w = exp(2*Pi*I/3);
F_4(x^4) = F(x)*F(I*x)*F(-x)*F(-I*x) = 1/(1-7*x^4+x^8);
F_5(x^5) = 1/(1-11*x^5-x^10);
In general,
F_n(x^n) = 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
...
The logarithmic derivative of this sequence begins:
A203319 = [1,3,7,19,26,81,92,267,358,848,980,3061,3030,7976,...].
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(exp(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n)))),n)}
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=vector(n+1, i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))))); polcoeff(exp(x*Ser(vector(n+1, m, L[m]/m))), n)}
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{a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); polcoeff(A, n)}
A203321
L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n*k)*x^(n*k)/k ).
Original entry on oeis.org
1, 3, 7, 19, 26, 75, 78, 211, 241, 518, 463, 1447, 1002, 2558, 2612, 5715, 3928, 11901, 7316, 21574, 17031, 35159, 23047, 80575, 40951, 108488, 86911, 206638, 107823, 370220, 173725, 570803, 372181, 816496, 451883, 1723741, 665150, 2048982, 1404150, 3705366, 1530859, 5892479
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 75*x^6/6 +...
where
L(x) = x*exp(1*x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 +...) +
x^2/2*exp(3*x^2 + 7*x^4/2 + 12*x^6/3 + 15*x^8/4 + 18*x^10/5 +...) +
x^3/3*exp(4*x^3 + 12*x^6/2 + 13*x^9/3 + 28*x^12/4 + 24*x^15/5 +...) +
x^4/4*exp(7*x^4 + 15*x^8/2 + 28*x^12/3 + 31*x^16/4 + 42*x^20/5 +...) +
x^5/5*exp(6*x^5 + 18*x^10/2 + 24*x^15/3 + 42*x^20/4 + 31*x^25/5 +...) +
x^6/6*exp(12*x^6 + 28*x^12/2 + 39*x^18/3 + 60*x^24/4 + 72*x^30/5 +...) +
x^7/7*exp(8*x^7 + 24*x^14/2 + 32*x^21/3 + 56*x^28/4 + 48*x^35/5 +...) +
x^8/8*exp(15*x^8 + 31*x^16/2 + 60*x^24/3 + 63*x^32/4 + 90*x^40/5 +...) +...
...
Equivalently, L(x) = Sum_{n>=1} P_n(x^n) * x^n/n where
P_n(x) = exp( Sum_{k>=1} sigma(n*k)*x^k/k ), which begin:
P_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...;
P_2(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...;
P_3(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 + 533*x^6 +...;
P_4(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 + 3057*x^6 +...;
P_5(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + 2486*x^6 +...;
P_6(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 +...;
P_7(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 +...;
P_8(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 +...;
...
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{a(n)=local(L=vector(max(n,1), i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), sigma(m*k)*x^(m*k)/k)+x*O(x^n))))); if(n<1,0,L[n])}
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{a(n)=local(A=1+x+x*O(x^n),P=exp(sum(k=1,n,sigma(k)*x^k/k)+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(P, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}
Showing 1-3 of 3 results.
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