A203319
a(n) = n*Fibonacci(n) * Sum_{d|n} 1/(d*Fibonacci(d)).
Original entry on oeis.org
1, 3, 7, 19, 26, 81, 92, 267, 358, 848, 980, 3061, 3030, 7976, 11042, 25099, 27150, 78642, 79440, 219884, 270704, 584862, 659112, 1977909, 1950651, 4735370, 6204499, 14189096, 14912642, 43168586, 41734340, 110786987, 135815060, 290854380, 339428752, 953889058, 893839230
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
where
L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
here F(n) = Fibonacci(n) = A000045(n).
Equivalently,
L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
here Lucas(n) = A000032(n).
Exponentiation of the l.g.f. equals the g.f. of A203318:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...
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a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* Jean-François Alcover, Dec 23 2015 *)
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{a(n)=if(n<1,0, n*fibonacci(n)*sumdiv(n,d,1/(d*fibonacci(d))) )}
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(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=x); L=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))); n*polcoeff(L,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)))))); n*polcoeff(log(A), n)}
A203320
G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n*k) * x^(n*k)/k ) ).
Original entry on oeis.org
1, 1, 2, 4, 9, 16, 35, 61, 124, 222, 427, 749, 1434, 2493, 4585, 8032, 14511, 25096, 44791, 77019, 135435, 232002, 402957, 685582, 1181399, 1998168, 3410288, 5741978, 9726821, 16286497, 27409625, 45672026, 76378731, 126706567, 210690588, 347954716, 575685559, 946756712
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 35*x^6 + 61*x^7 +...
G.f.: A(x) = exp( 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 +...;
...
Also, P_n(x^n) = Product_{k=0..n-1} P(u^k*x) where u = n-th root of unity:
P_1(x) = P(x), the partition function;
P_2(x^2) = P(x)*P(-x);
P_3(x^3) = P(x)*P(u*x)*P(u^2*x) where u = exp(2*Pi*I/3);
P_4(x^4) = P(x)*P(I*x)*P(I^2*x)*P(I^3*x) where I^2 = -1;
...
The logarithmic derivative of this sequence begins:
A203321 = [1,3,7,19,26,75,78,211,241,518,463,1447,1002,2558,...].
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{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/m), sigma(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),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)))))); polcoeff(A, n)}
A203413
G.f.: exp( Sum_{n>=1} A203414(n)*x^n/n ) where A203414(n) = n*Pell(n)*Sum_{d|n} 1/(d*Pell(d)).
Original entry on oeis.org
1, 1, 3, 8, 25, 64, 200, 512, 1528, 4048, 11654, 30585, 88601, 231295, 651713, 1733011, 4814031, 12685230, 35225415, 92628772, 254268558, 672643614, 1826716115, 4814931851, 13086575526, 34391797265, 92637759753, 244294085952, 654813738224, 1720509596070, 4606408076053
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 64*x^5 + 200*x^6 + 512*x^7 +...
where
log(A(x)) = x/(1-2*x-x^2) + (x^2/2)/(1-6*x^2+x^4) + (x^3/3)/(1-14*x^3-x^6) + (x^4/4)/(1-34*x^4+x^8) +...+ (x^n/n)/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...
Equivalently, log(A(x)) = Sum_{n>=1} G_n(x^n) * x^n/n
where G_n(x) = exp( Sum_{k>=1} A002203(n*k)*x^k/k ), which begin:
G_1(x) = x*(1 + 2*x + 5*x^2 + 12*x^3 + 29*x^4 +...+ Pell(n+1)*x^n +...
G_2(x) = 1 + 6*x^2 + 35*x^4 + 204*x^6 +...+ Pell(2*n+2)/2*x^(2*n) +...
G_3(x) = 1 + 14*x^3 + 197*x^6 + 2772*x^9 +...+ Pell(3*n+3)/5*x^(3*n) +...
G_4(x) = 1 + 34*x^4 + 1155*x^8 + 39236*x^12 +...+ Pell(4*n+4)/12*x^(4*n) +...
G_5(x) = 1 + 82*x^5 + 6725*x^10 + 551532*x^15 +...+ Pell(5*n+5)/29*x^(5*n) +...
G_6(x) = 1 + 198*x^6 + 39203*x^12 + 7761996*x^18 +...+ Pell(6*n+6)/70*x^(6*n) +...
For n>=1, G_n(x) = 1/(1 - A002203(n)*x + (-1)^n*x^2),
where the companion Pell numbers (offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726 16238,...].
The logarithmic derivative of this sequence begins:
A203414 = [1,5,16,61,146,554,1184,4149,9457,29890,63152,...].
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/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
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{a(n)=local(A=1);A=exp(sum(m=1,n+1,x^m*Pell(m)*sumdiv(m, d, 1/(d*Pell(d))) +x*O(x^n)));polcoeff(A,n)}
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{a(n)=local(A=1);A=exp(sum(m=1,n+1,(x^m/m)/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))));polcoeff(A,n)}
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{a(n)=local(A=1);A=exp(sum(m=1,n+1,(x^m/m)*exp(sum(k=1,floor((n+1)/m),A002203(m*k)*x^(m*k)/k)+x*O(x^n))));polcoeff(A, n)}
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{a(n)=local(A=1+2*x+x*O(x^n),G=1/(1-2*x-x^2+x*O(x^n)));A=exp(sum(m=1,n+1,(x^m/m)*round(prod(k=0,m-1,subst(G,x,exp(2*Pi*I*k/m)*x+x*O(x^n))))));polcoeff(A, n)}
A203414
a(n) = n*Pell(n) * Sum_{d|n} 1/(d*Pell(d)) where Pell(n) = A000129(n).
Original entry on oeis.org
1, 5, 16, 61, 146, 554, 1184, 4149, 9457, 29890, 63152, 222850, 434994, 1414642, 3140576, 9575893, 19323714, 65160959, 125877072, 408744626, 865638272, 2563647322, 5176349664, 17476326546, 33019614771, 102921708050, 220209942688, 657218691722, 1292253982322
Offset: 1
L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 61*x^4/4 + 146*x^5/5 + 554*x^6/6 +...
where
L(x) = x*(1 + 2*x + 5*x^2 + 12*x^3 + 29*x^4 +...+ Pell(n+1)*x^n +...) +
x^2/2*(1 + 6*x^2 + 35*x^4 + 204*x^6 +...+ Pell(2*n+2)/2*x^(2*n) +...) +
x^3/3*(1 + 14*x^3 + 197*x^6 + 2772*x^9 +...+ Pell(3*n+3)/5*x^(3*n) +...) +
x^4/4*(1 + 34*x^4 + 1155*x^8 + 39236*x^12 +...+ Pell(4*n+4)/12*x^(4*n) +...) +
x^5/5*(1 + 82*x^5 + 6725*x^10 + 551532*x^15 +...+ Pell(5*n+5)/29*x^(5*n) +...) +
x^6/6*(1 + 198*x^6 + 39203*x^12 + 7761996*x^18 +...+ Pell(6*n+6)/70*x^(6*n) +...) +...
Equivalently,
L(x) = x/(1-2*x-x^2) + (x^2/2)/(1-6*x^2+x^4) + (x^3/3)/(1-14*x^3-x^6) + (x^4/4)/(1-34*x^4+x^8) +...+ (x^n/n)/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...
where A002203 is the companion Pell numbers.
Exponentiation of the l.g.f. equals the g.f. of A203413:
exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 64*x^5 + 200*x^6 + 512*x^7 + 1528*x^8 + 4048*x^9 +...+ A203413(n)*x^n +...
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pell[n_] := pell[n] = ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2]) // Round; a[n_] := n pell[n] DivisorSum[n, 1/(# pell[#]) &]; Array[a, 30] (* Jean-François Alcover, Dec 23 2015 *)
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/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
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{a(n)=if(n<1, 0, n*Pell(n)*sumdiv(n, d, 1/(d*Pell(d))) )}
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{a(n)=n*polcoeff(sum(m=1, n+1, (x^m/m)/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
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{a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), A002203(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L, n)}
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{a(n)=local(A=1+2*x+x*O(x^n), F=1/(1-2*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)))))); n*polcoeff(log(A), n)}
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