A322209
L.g.f.: log( Product_{n>=1} 1/(1 - (2^n+1)*x^n) ).
Original entry on oeis.org
0, 3, 19, 54, 199, 408, 1612, 3090, 11023, 26487, 80994, 199686, 676540, 1700832, 5285096, 15197274, 45739039, 131368404, 401655943, 1172222958, 3549402474, 10533769146, 31617172980, 94336116834, 283990486780, 848323147233, 2546924693306, 7631598676410, 22903854049016, 68645946621360, 206035134959112, 617739968277066, 1853594327953471
Offset: 0
L.g.f.: L(x) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ...
such that
exp( L(x) ) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ... + A322199(n)*x^n + ...
also,
exp( L(x) ) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).
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{L = sum(n=1,41, -log(1 - (x^n + y^n) +O(x^41) +O(y^41)) );}
{A322200(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
{a(n) = sum(k=0,n, A322200(n-k,k)*2^k )}
for(n=0,40, print1( a(n),", ") )
A382976
Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).
Original entry on oeis.org
1, 3, 5, 24, 44, 129, 384, 897, 2220, 5706, 15268, 35178, 89829, 212982, 526222, 1294263, 3087570, 7300896, 17726100, 41705904, 98782950, 236059794, 551697495, 1293417672, 3033232130, 7081297146, 16430673765, 38347412562, 88762751808, 204970377366, 473719894598
Offset: 0
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n=30; CoefficientList[Normal@Series[Product[1+(2^k+1) x^k,{k,1,n}],{x,0,n}],x] (* Vincenzo Librandi, Apr 11 2025 *)
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f(n) = -1;
g(n) = -(2^n+1);
a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;
A382977
Expansion of Product_{k>=1} 1/(1 - (2^k - 1) * x^k).
Original entry on oeis.org
1, 1, 4, 11, 35, 87, 271, 659, 1908, 4832, 13132, 32688, 89109, 218385, 571489, 1427388, 3652877, 8980805, 22858201, 55822728, 140065621, 342001192, 845707856, 2052802367, 5057431745, 12197383588, 29738238996, 71604414162, 173406091548, 415167136507, 1000881376700
Offset: 0
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n := 30; R := PowerSeriesRing(Rationals(), n+1); f := &*[ 1 / (1 - (2^k - 1)*x^k) : k in [1..n] ]; coeffs := [Coefficient(f, i) : i in [0..n]]; coeffs; // Vincenzo Librandi, Apr 11 2025
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n=30; CoefficientList[Normal@Series[Product[1/(1-(2^k-1) x^k),{k,1,n}],{x,0,n}],x] (* Vincenzo Librandi, Apr 11 2025 *)
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f(n) = 1;
g(n) = 2^n-1;
a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;
A382978
Expansion of Product_{k>=1} (1 + (2^k - 1) * x^k).
Original entry on oeis.org
1, 1, 3, 10, 22, 67, 160, 433, 986, 2774, 6386, 16214, 39201, 95868, 229644, 569707, 1324730, 3186326, 7664378, 17955006, 42497434, 100710158, 235492595, 549267552, 1288847672, 2990756088, 6958113345, 16148883002, 37286262238, 85880711282, 198840926982, 454980392570
Offset: 0
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n := 31; R := PowerSeriesRing(Rationals(), n+1); f := &*[ (1 + (2^k - 1)*x^k) : k in [1..n] ]; coeffs := [Coefficient(f, i) : i in [0..n]];coeffs; // Vincenzo Librandi, Apr 11 2025
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n=31;CoefficientList[Normal@Series[Product[(1+(2^k-1) x^k),{k,1,n}],{x,0,n}],x] (* Vincenzo Librandi, Apr 11 2025 *)
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f(n) = -1;
g(n) = -(2^n-1);
a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;
Showing 1-4 of 4 results.
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