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L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n).
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9
1, 3, 1, 7, 1, 15, 1, 15, 10, 13, 1, 67, 1, 17, 16, 63, 1, 168, 1, 37, 22, 25, 1, 555, 1, 29, 118, 49, 1, 520, 1, 543, 34, 37, 1, 1048, 1, 41, 40, 1125, 1, 554, 1, 73, 475, 49, 1, 6651, 1, 563, 52, 85, 1, 1680, 1, 1457, 58, 61, 1, 20632, 1, 65, 787, 5087, 1
FORMULA
Forms the logarithmic derivative of A205478.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 +...
By definition:
L(x) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 +...
Exponentiation yields the g.f. of A205478: exp(L(x)) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 +...
MATHEMATICA
max = 70; s = Sum[(x^(n-1)/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}] + O[x]^max; CoefficientList[s, x]*Range[max] (* Jean-François Alcover, Dec 23 2015 *)
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+d*x^m+x*O(x^n))))), n)}
G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d) ).
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8
1, 1, 2, 3, 5, 8, 12, 20, 28, 45, 65, 101, 148, 221, 316, 469, 673, 969, 1420, 2025, 2892, 4100, 5905, 8314, 11860, 16645, 23399, 32838, 46071, 64274, 89761, 124977, 173231, 240492, 332978, 460015, 634271, 874464, 1200463, 1649499, 2263102, 3098661, 4239109
COMMENTS
Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^d) ) does not yield an integer series.
FORMULA
Logarithmic derivative yields A205477.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+2*x)*(1+x^2)/2 + x^3*(1+3*x)*(1+x^3)/3 + x^4*(1+4*x)*(1+2*x^2)*(1+x^4)/4 + x^5*(1+5*x)*(1+x^5)/5 + x^6*(1+6*x)*(1+3*x^2)*(1+2*x^3)*(1+x^6)/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 + 29*x^7/7 + 15*x^8/8 + 49*x^9/9 + 43*x^10/10 + ... + A205477(n)*x^n/n + ...
MATHEMATICA
max = 50; s = Exp[Sum[(x^n/n)*Product[1+n*x^d/d, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s , x] (* Jean-François Alcover, Dec 23 2015 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+m*x^d/d+x*O(x^n)))))), n)}
L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^(n/d))^d.
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8
1, 3, 7, 23, 76, 249, 974, 4151, 16558, 70308, 342937, 1680725, 8012252, 40903572, 222539812, 1202060807, 6608077855, 38523427818, 228629565951, 1349303611408, 8257330774574, 53118486147015, 345693735519287, 2252515985849693, 15028013765653626, 102689873016938288
FORMULA
Forms the logarithmic derivative of A205480.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 76*x^5/5 + 249*x^6/6 +...
By definition:
L(x) = x*(1+x) + x^2*(1+x^2)*(1+2*x)^2/2 + x^3*(1+x^3)*(1+3*x)^3/3 + x^4*(1+x^4)*(1+2*x^2)^2*(1+4*x)^4/4 + x^5*(1+x^5)*(1+5*x)^5/5 + x^6*(1+x^6)*(1+2*x^3)^2*(1+3*x^2)^3*(1+6*x)^6/6 +...
Exponentiation yields the g.f. of A205480: exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 76*x^6 + 242*x^7 +...
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^(m/d)+x*O(x^n))))), n)}
L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^n)^d.
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8
1, 3, 1, 11, 1, 45, 1, 59, 109, 53, 1, 869, 1, 101, 961, 3643, 1, 3555, 1, 18101, 3235, 245, 1, 92645, 21876, 341, 11287, 74141, 1, 722045, 1, 324667, 20329, 581, 502076, 5280611, 1, 725, 40054, 7567509, 1, 27239663, 1, 906301, 7838224, 1061, 1, 181474021
FORMULA
Forms the logarithmic derivative of A205482.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + x^3/3 + 11*x^4/4 + x^5/5 + 45*x^6/6 +...
By definition:
L(x) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)^2/2 + x^3*(1+x^3)*(1+3*x^3)^3/3 + x^4*(1+x^4)*(1+2*x^4)^2*(1+4*x^4)^4/4 + x^5*(1+x^5)*(1+5*x^5)^5/5 + x^6*(1+x^6)*(1+2*x^6)^2*(1+3*x^6)^3*(1+6*x^6)^6/6 +...
Exponentiation yields the g.f. of A205482: exp(L(x)) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 +...
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, d*log(1+d*x^m+x*O(x^n))))), n)}
L.g.f.: Sum_{n>=1} x^n/n * Product_{d|n} (1 + d*x^d)^n.
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8
1, 3, 4, 15, 31, 72, 176, 327, 751, 2063, 5138, 12708, 30993, 75386, 182644, 433255, 1004854, 2279349, 5115960, 11580835, 26533616, 62024966, 149683357, 373141332, 957942931, 2516465279, 6694846987, 17883365774, 47644695777, 125952933062, 329364348277
FORMULA
Forms the logarithmic derivative of A205484.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 31*x^5/5 + 72*x^6/6 +...
By definition:
L(x) = x*(1+x) + x^2*(1+x)^2*(1+2*x^2)^2/2 + x^3*(1+x)^3*(1+3*x^3)^3/3 + x^4*(1+x)^4*(1+2*x^2)^4*(1+4*x^4)^4/4 + x^5*(1+x)^5*(1+5*x^5)^5/5 + x^6*(1+x)^6*(1+2*x^2)^6*(1+3*x^3)^6*(1+6*x^6)^6/6 +...
Exponentiation yields the g.f. of A205484: exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 14*x^5 + 30*x^6 + 65*x^7 +...
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, m*log(1+d*x^d+x*O(x^n))))), n)}
L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^(n/d))^d.
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8
1, 3, 10, 43, 206, 1104, 6581, 43227, 307927, 2351288, 19124238, 165102052, 1507907818, 14512524085, 146581677005, 1548261405595, 17054944088112, 195518380169283, 2328512358930925, 28759349826041248, 367752208054445945, 4860792910118985370
FORMULA
Forms the logarithmic derivative of A205486.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 206*x^5/5 + 1104*x^6/6 +...
By definition:
L(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^2)^2*(1-4*x)^4) + (x^5/5)/((1-x^5)*(1-5*x)^5) + (x^6/6)/((1-x^6)*(1-2*x^3)^2*(1-3*x^2)^3*(1-6*x)^6) +...
Exponentiation yields the g.f. of A205486: exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 60*x^5 + 259*x^6 + 1273*x^7 +...
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^(m/d)+x*O(x^n))))), n)}
L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n)^d.
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8
1, 3, 4, 15, 6, 78, 8, 247, 202, 708, 12, 4146, 14, 5498, 8964, 24135, 18, 81114, 20, 206520, 193736, 225558, 24, 2314378, 242656, 1278332, 3622954, 9209950, 30, 26654118, 32, 58890983, 59213598, 35652216, 28736938, 628796418, 38, 179307278, 878319368
FORMULA
Forms the logarithmic derivative of A205488.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 15*x^4/4 + 6*x^5/5 + 78*x^6/6 +...
By definition:
L(x) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)^2) + (x^3/3)/((1-x^3)*(1-3*x^3)^3) + (x^4/4)/((1-x^4)*(1-2*x^4)^2*(1-4*x^4)^4) + (x^5/5)/((1-x^5)*(1-5*x^5)^5) + (x^6/6)/((1-x^6)*(1-2*x^6)^2*(1-3*x^6)^3*(1-6*x^6)^6) +...
Exponentiation yields the g.f. of A205488: exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 26*x^6 + 32*x^7 +...
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -d*log(1-d*x^m+x*O(x^n))))), n)}
L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^d)^n.
+10
8
1, 3, 7, 23, 51, 165, 386, 1039, 2554, 6963, 17260, 47825, 124840, 340658, 911037, 2484687, 6614616, 17735646, 46647167, 122536323, 318125129, 825153684, 2130076369, 5522611009, 14375957026, 37817347272, 100579846732, 271246531726, 740731197176
FORMULA
Forms the logarithmic derivative of A205490.
EXAMPLE
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 165*x^6/6 +...
By definition:
L(x) = x/(1-x) + (x^2/2)/((1-x)^2*(1-2*x^2)^2) + (x^3/3)/((1-x)^3*(1-3*x^3)^3) + (x^4/4)/((1-x)^4*(1-2*x^2)^4*(1-4*x^4)^4) + (x^5/5)/((1-x)^5*(1-5*x^5)^5) + (x^6/6)/((1-x)^6*(1-2*x^2)^6*(1-3*x^3)^6*(1-6*x^6)^6) +...
Exponentiation yields the g.f. of A205490: exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 57*x^6 + 134*x^7 +...
PROG
(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -m*log(1-d*x^d+x*O(x^n))))), n)}
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