A322204 -id:A322204 - cp's OEIS frontend

A322203 a(n) = coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)), for n >= 1.

1, 5, 13, 45, 131, 497, 1723, 6525, 24349, 92655, 352727, 1353177, 5200313, 20061767, 77559203, 300553245, 1166803127, 4537617761, 17672631919, 68923449895, 269128942459, 1052050187347, 4116715363823, 16123804567209, 63205303219531, 247959276874717, 973469712897103, 3824345340503999, 15033633249770549, 59132290937828607, 232714176627630575, 916312071072401757

Offset: 1

Paul D. Hanna, Nov 30 2018

nonn

			G.f.: L(x) = x + 5*x^2/2 + 13*x^3/3 + 45*x^4/4 + 131*x^5/5 + 497*x^6/6 + 1723*x^7/7 + 6525*x^8/8 + 24349*x^9/9 + 92655*x^10/10 + 352727*x^11/11 + 1353177*x^12/12 + ...
such that
exp( L(x) ) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 54*x^5 + 168*x^6 + 518*x^7 + 1702*x^8 + 5672*x^9 + 19413*x^10 + 67329*x^11 + ... + A322204(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A322204, A322202, A322200.

{L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{a(n) = polcoeff( n*polcoeff( L,n,x),n,y)}
for(n=1,35, print1( a(n),", ") )

a(n) = A322200(n,n)/2 for n >= 1.

a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 18 2019

A322202 G.f.: exp( Sum_{n>=1} A322201(n)x^n/n ), where A322201(n) is the coefficient of x^ny^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 2, 7, 20, 63, 190, 613, 1976, 6604, 22368, 77270, 270208, 956780, 3419212, 12323226, 44723840, 163320766, 599601984, 2211844684, 8193734760, 30469278673, 113692852342, 425558528235, 1597428832560, 6011972255226, 22680620270712, 85754229105470, 324898592591960, 1233299357981416, 4689870496585016, 17863799895741982, 68149300647823612, 260364494604701847, 996086232267182566, 3815683108118138847, 14634441964549504036

Offset: 0

Paul D. Hanna, Nov 30 2018

nonn

Conjecture: Euler transform of A123611. - Vaclav Kotesovec, Dec 12 2020

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 20*x^3 + 63*x^4 + 190*x^5 + 613*x^6 + 1976*x^7 + 6604*x^8 + 22368*x^9 + 77270*x^10 + 270208*x^11 + 956780*x^12 + ...
such that
log( A(x) ) = 2*x + 10*x^2/2 + 26*x^3/3 + 90*x^4/4 + 262*x^5/5 + 994*x^6/6 + 3446*x^7/7 + 13050*x^8/8 + 48698*x^9/9 + 185310*x^10/10 + ... + A322201(n)*x^n/n + ...
sqrt(A(x)) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 54*x^5 + 168*x^6 + 518*x^7 + 1702*x^8 + 5672*x^9 + 19413*x^10 + 67329*x^11 + 236994*x^12 + ... + A322204(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A322200, A322201, A322204.

nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 12 2020 *)
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 12 2020 *)

{L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{A322201(n) = polcoeff( 2*n*polcoeff( L,n,x),n,y)}
{a(n) = polcoeff( exp( sum(m=1,n, A322201(m)*x^m/m ) +x*O(x^n) ),n) }
for(n=0,35, print1( a(n),", ") )

a(n) ~ c * 4^n / n^(3/2), where c = 4/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1)))^2 = 2.704933139869066452954644773467... - Vaclav Kotesovec, Jun 18 2019, updated Dec 12 2020

G.f.: Product_{j>=1} c(x^j)^2, where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - Vaclav Kotesovec, Dec 12 2020

A327683 Expansion of Product_{k>0} (1+sqrt(1+4*x^k))/2.

Original entry on oeis.org

1, 1, 0, 4, -5, 17, -40, 144, -459, 1517, -5111, 17747, -62074, 219292, -782602, 2816664, -10205754, 37203230, -136360106, 502219652, -1857659296, 6897983144, -25704335380, 96090440940, -360265425619, 1354343161419, -5103948546609, 19278502980063, -72972099256954

Offset: 0

Seiichi Manyama, Sep 22 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Convolution inverse of A327682.

Cf. A000009, A268498, A322204, A327684.

N:= 40:
P:= mul((1+sqrt(1+4*x^k))/2,k=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Sep 22 2019

nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1+4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 22 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1+4*x^k))/2))

N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, (-1)^j*binomial(2*j-2, j-1)*x^(i*j)/j)))

a(n) ~ -(-1)^n * c * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + sqrt(1 + 4*(-1/4)^k))/2 = 0.52271977595412566689522667777276363119313248923... - Vaclav Kotesovec, May 06 2021

A322188 G.f.: exp( Sum_{n>=1} A322187(n)x^n/n ), where A322187(n) is the coefficient of x^ny^n/n in log( Product_{n>=1} 1/(1 - x^(2n-1) - y^(2n-1)) ).

Original entry on oeis.org

1, 1, 2, 6, 15, 45, 140, 448, 1483, 5027, 17311, 60469, 213678, 762284, 2741864, 9932346, 36202666, 132677658, 488605698, 1807176452, 6710206574, 25003642942, 93468147306, 350425771854, 1317330452697, 4964398631867, 18751217069083, 70975750129731, 269180061675328, 1022750160098864, 3892577330120307, 14838784128136803, 56651259287153670, 216586672901518164, 829142137823283601, 3178107527615273349

Offset: 0

Paul D. Hanna, Dec 07 2018

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 45*x^5 + 140*x^6 + 448*x^7 + 1483*x^8 + 5027*x^9 + 17311*x^10 + 60469*x^11 + 213678*x^12 + ...
such that
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 35*x^4/4 + 131*x^5/5 + 471*x^6/6 + 1723*x^7/7 + 6435*x^8/8 + 24349*x^9/9 + 92393*x^10/10 + 352727*x^11/11 + 1352183*x^12/12 + ... + A322187(n)*x^n/n + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 46*x^4 + 144*x^5 + 466*x^6 + 1536*x^7 + 5187*x^8 + 17842*x^9 + 62209*x^10 + 219504*x^11 + 782272*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..512

Cf. A322187, A322198, A322204.

N=35;
{L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }
{A322187(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1, n, A322187(m)*x^m/m ) +x*O(x^n) ), n) }
for(n=0, N, print1( a(n), ", ") )

a(n) ~ c * 4^n / n^(3/2), where c = 0.57389010009720382786456367148681469430628117317... - Vaclav Kotesovec, Jun 18 2019

A322207 a(n) = coefficient of x^(3n)y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 9, 58, 473, 3881, 33786, 296017, 2630521, 23535994, 211922929, 1917334794, 17417202554, 158753389913, 1451183583033, 13298522310098, 122131739530937, 1123787895356429, 10358022488568858, 95615237915961119, 883829035976891713, 8179808679273553156, 75788358479315971850, 702916267465270526873, 6525429588311530420858, 60629817430084280273281

Offset: 1

Paul D. Hanna, Dec 01 2018

nonn

			G.f.: L(x) = x + 9*x^2/2 + 58*x^3/3 + 473*x^4/4 + 3881*x^5/5 + 33786*x^6/6 + 296017*x^7/7 + 2630521*x^8/8 + 23535994*x^9/9 + 211922929*x^10/10 + 1917334794*x^11/11 + 17417202554*x^12/12 + ...
such that
exp( L(x) ) = 1 + x + 5*x^2 + 24*x^3 + 150*x^4 + 1002*x^5 + 7296*x^6 + 55082*x^7 + 429803*x^8 + 3429141*x^9 + 27861573*x^10 + 229668027*x^11 + 1916090676*x^12 + ... + A322208(n)*x^n + ...

Cf. A322200, A322208, A322206, A322204.

{L = sum(n=1,121, -log(1 - (x^n + y^n) +O(x^121) +O(y^121)) );}
{a(n) = polcoeff( n*polcoeff( L,3*n,x),n,y)}
for(n=1,40, print1( a(n),", ") )

a(n) = A322200(3*n,n)/4.

Logarithmic derivative of A322208.

A327682 Expansion of Product_{k>0} (-1+sqrt(1+4x^k))/(2x^k).

Original entry on oeis.org

1, -1, 1, -5, 14, -40, 122, -404, 1362, -4608, 15881, -55709, 197402, -705114, 2539282, -9210196, 33605471, -123262137, 454268676, -1681305246, 6246544735, -23288217459, 87096982499, -326680267261, 1228547420236, -4631474743422, 17499462106763, -66257720483935, 251356773101419

Offset: 0

Seiichi Manyama, Sep 22 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A081362, A168491, A309867, A322204, A327683.

m = 28; CoefficientList[Series[Product[(-1 + Sqrt[1 + 4*x^k])/(2*x^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 06 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (-1+sqrt(1+4*x^k))/(2*x^k)))

N=66; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, N\i, (-1)^j*binomial(2*j, j)*x^(i*j)/(j+1))))

a(n) ~ (-1)^n * c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (-1 + sqrt(1 + 4*(-1/4)^k)) / (2*(-1/4)^k) = 0.5396673413761086071059510679780476790558662471136055... - Vaclav Kotesovec, May 06 2021

A309682 G.f.: C(x)C(2x^2)C(3x^3)*..., where C(x) is the g.f. for A000108.

Original entry on oeis.org

1, 1, 4, 10, 33, 81, 282, 762, 2599, 7979, 27343, 89371, 315256, 1078498, 3857048, 13651786, 49475282, 178736186, 655247192, 2401663838, 8883371016, 32906649488, 122619768860, 457836275272, 1716620421629, 6449729802639, 24308647131627, 91800114425437

Offset: 0

Vaclav Kotesovec, Aug 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1669

Cf. A025174, A110143, A322204, A326985.

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1,
      C(n), add(C(j)*i^j*b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019

nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*k*x^k])/(2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (2^k*(2^(k-1) - sqrt(4^(k-1) - k))/k) = 0.711438694828613555153724789...

A309867 Expansion of Product_{k>0} (1+sqrt(1-4*x^k))/2.

Original entry on oeis.org

1, -1, -2, -2, -5, -9, -36, -104, -365, -1219, -4213, -14617, -51570, -183084, -656536, -2370066, -8613590, -31478538, -115632718, -426676244, -1580878746, -5878933054, -21936060630, -82100980070, -308146839623, -1159545407027, -4373730398473, -16533813947503

Offset: 0

Seiichi Manyama, Sep 22 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Convolution inverse of A322204.

Cf. A115140, A327682, A327683.

nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1-4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1-4*x^k))/2))

N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, binomial(2*j-2, j-1)*x^(i*j)/j)))

a(n) ~ -c * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + sqrt(1 - 4*(1/4)^k))/2 = 0.4567034206737725013365271429022657551331606541289778092649... - Vaclav Kotesovec, May 06 2021

A322186 G.f.: exp( Sum_{n>=1} A322185(n)x^n/n ), where A322185(n) = sigma(2n) * binomial(2*n,n)/2.

Original entry on oeis.org

1, 3, 15, 76, 357, 1662, 8203, 36609, 169800, 788024, 3586350, 15948147, 73761986, 324147729, 1454796651, 6544916640, 28902107643, 126842754933, 567156315794, 2468434955040, 10893525305088, 47854663427104, 208582052412240, 905923236202737, 3975385018556868, 17200981327476354, 74619131550054048, 323976744392754994, 1400917964875907424, 6031485491299656747

Offset: 0

Paul D. Hanna, Dec 07 2018

nonn

Related series:
(1) Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 = exp( Sum_{n>=1} sigma(2*n) * x^n/n ) (see formula of Joerg Arndt in A182818).
(2) C(x) = exp( Sum_{n>=1} binomial(2*n,n)/2 * x^n/n ), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
A322185(n) is also the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - (x + y)^n) ).

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 76*x^3 + 357*x^4 + 1662*x^5 + 8203*x^6 + 36609*x^7 + 169800*x^8 + 788024*x^9 + 3586350*x^10 + 15948147*x^11 + ...
such that
log(A(x)) = 3*x + 21*x^2/2 + 120*x^3/3 + 525*x^4/4 + 2268*x^5/5 + 12936*x^6/6 + 41184*x^7/7 + 199485*x^8/8 + 948090*x^9/9 + 3879876*x^10/10 + 12697776*x^11/11 + ... + A322185(n)*x^n/n + ...
RELATED SERIES.
A(x)^2 = 1 + 6*x + 39*x^2 + 242*x^3 + 1395*x^4 + 7746*x^5 + 42864*x^6 + 226560*x^7 + 1185417*x^8 + 6126642*x^9 + 31178598*x^10 + 156270312*x^11 + 780797727*x^12 + ...
where A(x)^2 = exp( Sum_{n>=1} sigma(2*n) * binomial(2*n,n) * x^n/n ).

Paul D. Hanna, Table of n, a(n) for n = 0..512

Cf. A322185, A322204, A322188.

{A322185(n) = sigma(2*n) * binomial(2*n,n)/2}
{a(n) = polcoeff( exp( sum(m=1, n, A322185(m)*x^m/m ) +x*O(x^n) ), n) }
for(n=0, 30, print1( a(n), ", ") )

A322206 G.f.: exp( Sum_{n>=1} A322205(n)x^n/n ), where A322205(n) is the coefficient of x^(2n)*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 1, 4, 14, 63, 294, 1526, 8157, 45332, 257378, 1489539, 8744722, 51965701, 311915649, 1888382937, 11517313486, 70699038868, 436454255701, 2708000234769, 16877547822830, 105614312726477, 663314865710063, 4179789872458354, 26418030929753007, 167435388627981690, 1063892712455899336, 6775891814778961392, 43249097401730644817, 276606084622479837727, 1772391802339441687335, 11376702892986621823617

Offset: 0

Paul D. Hanna, Dec 01 2018

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 63*x^4 + 294*x^5 + 1526*x^6 + 8157*x^7 + 45332*x^8 + 257378*x^9 + 1489539*x^10 + 8744722*x^11 + 51965701*x^12 + ...
such that
log( A(x) ) = x + 7*x^2/2 + 31*x^3/3 + 179*x^4/4 + 1006*x^5/5 + 6265*x^6/6 + 38767*x^7/7 + 245515*x^8/8 + 1562368*x^9/9 + 10017042*x^10/10 + ... + A322205(n)*x^n/n + ...
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2 + 67*x^3 + 333*x^4 + 1686*x^5 + 9031*x^6 + 49629*x^7 + 280467*x^8 + 1614932*x^9 + 9449961*x^10 + 56001366*x^11 + 335437797*x^12 + ...

Cf. A322200, A322205, A322204.

{L = sum(n=1,81, -log(1 - (x^n + y^n) +O(x^81) +O(y^81)) );}
{A322205(n) = polcoeff( n*polcoeff( L,2*n,x),n,y)}
{a(n) = polcoeff( exp( sum(m=1,n, A322205(m)*x^m/m ) +x*O(x^n) ),n) }
for(n=0,40, print1( a(n),", ") )

cp's OEIS Frontend

A322203 a(n) = coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)), for n >= 1.

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A322202 G.f.: exp( Sum_{n>=1} A322201(n)*x^n/n ), where A322201(n) is the coefficient of x^n*y^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

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A327683 Expansion of Product_{k>0} (1+sqrt(1+4*x^k))/2.

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A322188 G.f.: exp( Sum_{n>=1} A322187(n)*x^n/n ), where A322187(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ).

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A322207 a(n) = coefficient of x^(3*n)*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

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A327682 Expansion of Product_{k>0} (-1+sqrt(1+4*x^k))/(2*x^k).

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A309682 G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.

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A309867 Expansion of Product_{k>0} (1+sqrt(1-4*x^k))/2.

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A322202 G.f.: exp( Sum_{n>=1} A322201(n)x^n/n ), where A322201(n) is the coefficient of x^ny^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

A322188 G.f.: exp( Sum_{n>=1} A322187(n)x^n/n ), where A322187(n) is the coefficient of x^ny^n/n in log( Product_{n>=1} 1/(1 - x^(2n-1) - y^(2n-1)) ).

A322207 a(n) = coefficient of x^(3n)y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

A327682 Expansion of Product_{k>0} (-1+sqrt(1+4x^k))/(2x^k).

A309682 G.f.: C(x)C(2x^2)C(3x^3)*..., where C(x) is the g.f. for A000108.

A322186 G.f.: exp( Sum_{n>=1} A322185(n)x^n/n ), where A322185(n) = sigma(2n) * binomial(2*n,n)/2.

A322206 G.f.: exp( Sum_{n>=1} A322205(n)x^n/n ), where A322205(n) is the coefficient of x^(2n)*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).