A182818 -id:A182818 - cp's OEIS frontend

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

1, 8, 44, 192, 726, 2464, 7704, 22527, 62329, 164516, 416948, 1019690, 2416246, 5565864, 12498215, 27421815, 58903768, 124088548, 256749822, 522450250, 1046735092, 2066948472, 4026431543, 7743987036, 14715788745, 27648250012, 51390298666, 94550761844

Offset: 0

Seiichi Manyama, Feb 28 2017

nonn

			G.f.: A(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + ...
log(A(x)) = 8*x + 24*x^2/2 + 32*x^3/3 + 56*x^4/4 + 48*x^5/5 + 96*x^6/6 + 57*x^7/7 + 120*x^8/8 + ... + sigma(7*n)*x^n/n + ...

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

Cf. A282942 (Product_{n>=1} (1 - x^n)^8/(1 - x^(7*n))), A283078 (sigma(7*n)).

Cf. exp( Sum_{n>=1} sigma(k*n)*x^n/n ): A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), this sequence (k=7), A283120 (k=8), A283121 (k=9).

G.f.: exp( Sum_{n>=1} sigma(7*n)*x^n/n ).
a(n) = (1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 3025 * exp(sqrt(110*n/21)*Pi) / (28224*sqrt(14)*n^(5/2)). - Vaclav Kotesovec, Mar 20 2017

A283224 Expansion of exp( Sum_{n>=1} sigma_2(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 5, 23, 90, 317, 1036, 3192, 9358, 26336, 71542, 188440, 483007, 1208275, 2956941, 7093531, 16709523, 38706389, 88281394, 198474497, 440263342, 964424210, 2087882510, 4470194335, 9471079495, 19868723042, 41291454537, 85049747913, 173697766646

Offset: 0

Seiichi Manyama, Mar 03 2017

nonn

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

Cf. exp( Sum_{n>=1} sigma_k(2*n)*x^n/n ): A182818 (k=1), this sequence (k=2).

Cf. exp( Sum_{n>=1} sigma_2(m*n)*x^n/n ): A000219 (m=1), this sequence (m=2), A283238 (m=3).

a(n) = (1/n)*Sum_{k=1..n} sigma_2(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A203320 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(nk) 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

Paul D. Hanna, Dec 31 2011

nonn

			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,...].

Paul D. Hanna, Table of n, a(n) for n = 0..120

Cf. A203321; A000041, A182818, A182819, A182820, A182821.

{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)}

{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)}

G.f.: exp( Sum_{n>=1} P_n(x^n) * x^n/n ) where P_n(x^n) = Product_{k=0..n-1} P(u^k*x), u is an n-th root of unity, and P(x) is the partition function (A000041); P(x) = exp(Sum_{n>=1} sigma(n)*x^n/n) where sigma(n) is the sum of divisors of n (A000203).

The logarithmic derivative yields A203321.

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 9, 77, 534, 3320, 18933, 100770, 506697, 2428161, 11161765, 49469005, 212246744, 884491121, 3589900607, 14223638534, 55122970206, 209307080221, 779837798559, 2854660220661, 10278494869342, 36439277959593, 127311828611819, 438712861233581

Offset: 0

Seiichi Manyama, Mar 03 2017

nonn

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

Cf. exp( Sum_{n>=1} sigma_k(2*n)*x^n/n ): A182818 (k=1), A283224 (k=2), this sequence (k=3).

Cf. exp( Sum_{n>=1} sigma_3(m*n)*x^n/n ): A023871 (m=1), this sequence (m=2), A283244 (m=3).

a(n) = (1/n)*Sum_{k=1..n} sigma_3(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A319362 a(n) = [x^n] exp(Sum_{k>=1} sigma(nk)x^k/k).

Original entry on oeis.org

1, 1, 8, 39, 385, 917, 31247, 22527, 1081986, 2464860, 50099635, 14931071, 19684696065, 394805109, 82267017929, 496514888157, 11386442827781, 284625019799, 3469798073972537, 7725084195239, 136470024990370842, 28400489198168457, 241211623942678951, 5776331152550399

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

Seiichi Manyama, Table of n, a(n) for n = 0..959

Cf. A000041, A182818, A182819, A182820, A182821, A283077, A283119, A283120, A283121, A319363.

Table[SeriesCoefficient[Exp[Sum[DivisorSigma[1, n k] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 23}]

A322185 a(n) = sigma(2n) binomial(2*n,n)/2, for n >= 1.

Original entry on oeis.org

3, 21, 120, 525, 2268, 12936, 41184, 199485, 948090, 3879876, 12697776, 81124680, 218412600, 1123264800, 5584230720, 18934032285, 63007367940, 412918656150, 1060357914000, 6203093796900, 25836377973120, 88372156476240, 296403506193600, 1999351428352200, 5878093199355468, 24300008114457096, 116816365538886720, 458921436045626400, 1353026992479346800

Offset: 1

Paul D. Hanna, Dec 07 2018

nonn

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

			L.g.f: L(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 + ... + sigma(2*n) * binomial(2*n,n)/2 * x^n/n + ...
RELATED SERIES.
exp(L(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 + ... + A322186(n)*x^n + ...
The table of coefficients of x^n*y^k/(n+k) in
log( Product_{n>=1} 1/(1 - (x + y)^n) ) = (1*x + 1*y)/1 + (3*x^2 + 6*x*y + 3*y^2)/2 + (4*x^3 + 12*x^2*y + 12*x*y^2 + 4*y^3)/3 + (7*x^4 + 28*x^3*y + 42*x^2*y^2 + 28*x*y^3 + 7*y^4)/4 + (6*x^5 + 30*x^4*y + 60*x^3*y^2 + 60*x^2*y^3 + 30*x*y^4 + 6*y^5)/5 + (12*x^6 + 72*x^5*y + 180*x^4*y^2 + 240*x^3*y^3 + 180*x^2*y^4 + 72*x*y^5 + 12*y^6)/6 + (8*x^7 + 56*x^6*y + 168*x^5*y^2 + 280*x^4*y^3 + 280*x^3*y^4 + 168*x^2*y^5 + 56*x*y^6 + 8*y^7)/7 + (15*x^8 + 120*x^7*y + 420*x^6*y^2 + 840*x^5*y^3 + 1050*x^4*y^4 + 840*x^3*y^5 + 420*x^2*y^6 + 120*x*y^7 + 15*y^8)/8 + ...
begins
n=0: [0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, ..., sigma(k), ...];
n=1: [1, 6, 12, 28, 30, 72, 56, 120, 117, 180, ...];
n=2: [3, 12, 42, 60, 180, 168, 420, 468, 810, 660, ...];
n=3: [4, 28, 60, 240, 280, 840, 1092, 2160, 1980, 6160, ...];
n=4: [7, 30, 180, 280, 1050, 1638, 3780, 3960, 13860, 10010, ...];
n=5: [6, 72, 168, 840, 1638, 4536, 5544, 22176, 18018, 48048, ...];
n=6: [12, 56, 420, 1092, 3780, 5544, 25872, 24024, 72072, 120120, ...];
n=7: [8, 120, 468, 2160, 3960, 22176, 24024, 82368, 154440, 354640, ...];
n=8: [15, 117, 810, 1980, 13860, 18018, 72072, 154440, 398970, 437580, ...];
n=9: [13, 180, 660, 6160, 10010, 48048, 120120, 354640, 437580, 1896180, ...];
n=10: [18, 132, 1848, 4004, 24024, 72072, 248248, 350064, 1706562, 1847560, ...]; ...
in which the diagonal of coefficients of x^n*y^n/(2*n) equals
[0, 6, 42, 240, 1050, 4536, 25872, 82368, 398970, 1896180, ..., 2*a(n), ...],
which is twice this sequence.

Paul D. Hanna, Table of n, a(n) for n = 1..512

Cf. A322186, A322203, A322187.

{a(n) = sigma(2*n) * binomial(2*n,n)/2}
for(n=1, 30, print1( a(n), ", ") )

/* [x^n*y^n/n] log( Product_{n>=1} 1/(1 - (x + y)^n) ) */
N=30
{L = sum(n=1, 2*N+1, -log(1 - (x + y)^n +x*O(x^(2*N)) +y*O(y^(2*N))) ); }
{a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
for(n=1, N, print1( a(n), ", ") )

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

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), ", ") )

cp's OEIS Frontend

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

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A283224 Expansion of exp( Sum_{n>=1} sigma_2(2*n)*x^n/n ) in powers of x.

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A203320 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n*k) * x^(n*k)/k ) ).

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A282327 Expansion of exp( Sum_{n>=1} sigma_3(2*n)*x^n/n ) in powers of x.

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A319362 a(n) = [x^n] exp(Sum_{k>=1} sigma(n*k)*x^k/k).

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A322185 a(n) = sigma(2*n) * binomial(2*n,n)/2, for n >= 1.

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

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A283224 Expansion of exp( Sum_{n>=1} sigma_2(2n)x^n/n ) in powers of x.

A203320 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(nk) x^(n*k)/k ) ).

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

A319362 a(n) = [x^n] exp(Sum_{k>=1} sigma(nk)x^k/k).

A322185 a(n) = sigma(2n) binomial(2*n,n)/2, for n >= 1.

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