cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A158112 a(n) = [x^n] eta(x)^(2^n).

Original entry on oeis.org

1, -2, 2, 0, 260, -72384, 40618368, -64586112000, 322969883412000, -5357637135507147264, 305435344239722874022912, -61394251001333873555321724928, 44347756583930343994689166159720448
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 - 2*x + 2*x^2 + 260*x^4 - 72384*x^5 +...
A(x) = 1 + log(eta(2*x)) + log(eta(4*x))^2/2! + log(eta(8*x))^3/3! +...
...
Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
then a(n) is the coefficient of x^n in eta(x)^(2^n):
eta(x)^(2^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..];
eta(x)^(2^1): [1,(-2),-1,2,1,2,-2,0,-2,-2,1,0,0,2,3,-2,2,...];
eta(x)^(2^2): [1,-4,(2),8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...];
eta(x)^(2^3): [1,-8,20,(0),-70,64,56,0,-125,-160,308,0,110,0,...];
eta(x)^(2^4): [1,-16,104,-320,(260),1248,-3712,1664,6890,...];
eta(x)^(2^5): [1,-32,464,-3968,21576,(-72384),109120,215296,...];
eta(x)^(2^6): [1,-64,1952,-37632,512400,-5207936,(40618368),...]; ...
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A010815, A158113, A158114, A158115, A158102, A158103, A158104, A158105.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[QPochhammer[q]^(2^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *)
  • PARI
    {a(n)=polcoeff(eta(x+x*O(x^n))^(2^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,log(eta(2^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(2^m*x)^k/(1-(2^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} log( eta(2^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (2^n*x)^k/(1 - (2^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] Product_{k>=1} (1-x^k)^(2^n).

A158102 a(n) = [x^n] 1/eta(x)^(2^n).

Original entry on oeis.org

1, 2, 14, 192, 6460, 604352, 176638592, 175532923904, 624721970104800, 8175475971987294720, 397644417676499319116800, 72130520531109535961648537600, 48849638890950023005189368829890560
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 192*x^3 + 6460*x^4 + 604352*x^5 +...
Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
then a(n) is the coefficient of x^n in P(x)^(2^n):
P(x)^(2^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
P(x)^(2^1): [1,(2),5,10,20,36,65,110,185,300,481,752,1165,...];
P(x)^(2^2): [1,4,(14),40,105,252,574,1240,2580,5180,10108,...];
P(x)^(2^3): [1,8,44,(192),726,2464,7704,22528,62337,164560,...];
P(x)^(2^4): [1,16,152,1088,(6460),33440,155584,663936,...];
P(x)^(2^5): [1,32,560,7040,70840,(604352),4528832,30529280,...];
P(x)^(2^6): [1,64,2144,49920,905840,13627264,(176638592),...]; ...
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A000041, A158103, A158104, A158105, A158112, A158113, A158114, A158115.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[1/QPochhammer[q]^(2^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *)
  • PARI
    {a(n)=polcoeff(1/eta(x+x*O(x^n))^(2^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(2^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(2^m*x)^k/(1-(2^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(2^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (2^n*x)^k/(1 - (2^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] P(x)^(2^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).

A158103 a(n) = [x^n] 1/eta(x)^(3^n).

Original entry on oeis.org

1, 3, 54, 4410, 2208465, 7958364696, 221555929999779, 48859965926267395185, 86255750314864791590005098, 1228682270675324224826503933533795, 142349199783036538823503393789360721783250
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 + 3*x + 54*x^2 + 4410*x^3 + 2208465*x^4 +...
A(x) = 1 - log(eta(3*x)) + log(eta(9*x))^2/2! - log(eta(27*x))^3/3! +-...
...
Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
then a(n) is the coefficient of x^n in P(x)^(3^n):
P(x)^(3^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
P(x)^(3^1): [1,(3),9,22,51,108,221,429,810,1479,2640,4599,...];
P(x)^(3^2): [1,9,(54),255,1035,3753,12483,38709,113265,...];
P(x)^(3^3): [1,27,405,(4410),38745,290466,1923075,11506185,...];
P(x)^(3^4): [1,81,3402,98523,(2208465),40795083,645824907,...];
P(x)^(3^5): [1,243,29889,2480382,156189951,(7958364696),...];
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A000041, A158102, A158104, A158105, A158112, A158113, A158114, A158115.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[1/QPochhammer[q]^(3^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *)
  • PARI
    {a(n)=polcoeff(1/eta(x+x*O(x^n))^(3^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(3^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(3^m*x)^k/(1-(3^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(3^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (3^n*x)^k/(1 - (3^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] P(x)^(3^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).

A158104 a(n) = [x^n] 1/eta(x)^(4^n).

Original entry on oeis.org

1, 4, 152, 49920, 191701440, 9659304851456, 6631121047328399360, 63121566305846614746333184, 8450364710466604643357631528951808, 16116664948206611884835318695391740990586880
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 + 4*x + 152*x^2 + 49920*x^3 + 191701440*x^4 +...
A(x) = 1 - log(eta(4*x)) + log(eta(16*x))^2/2! - log(eta(64*x))^3/3! +-...
...
Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
then a(n) is the coefficient of x^n in P(x)^(4^n):
P(x)^(4^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
P(x)^(4^1): [1,(4),14,40,105,252,574,1240,2580,5180,10108,...];
P(x)^(4^2): [1,16,(152),1088,6460,33440,155584,663936,2636326,...];
P(x)^(4^3): [1,64,2144,(49920),905840,13627264,176638592,...];
P(x)^(4^4): [1,256,33152,2894848,(191701440),10266643968,...];
P(x)^(4^5): [1,1024,525824,180531200,46620870400,(9659304851456),...];
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A000041, A158102, A158103, A158105, A158112, A158113, A158114, A158115.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[1/QPochhammer[q]^(4^n), {q, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Nov 24 2015 *)
  • PARI
    {a(n)=polcoeff(1/eta(x+x*O(x^n))^(4^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(4^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(4^m*x)^k/(1-(4^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(4^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (4^n*x)^k/(1 - (4^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] P(x)^(4^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).

A158105 a(n) = [x^n] 1/eta(x)^(5^n).

Original entry on oeis.org

1, 5, 350, 349125, 6541895625, 2507423437503750, 20269247819848681671875, 3527360416889444660042550312500, 13447858828542523681813067677875519531250
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 + 5*x + 350*x^2 + 349125*x^3 + 6541895625*x^4 +...
A(x) = 1 - log(eta(5*x)) + log(eta(25*x))^2/2! - log(eta(125*x))^3/3! +-...
...
Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
then a(n) is the coefficient of x^n in P(x)^(5^n):
P(x)^(5^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
P(x)^(5^1): [1,(5),20,65,190,506,1265,2990,6765,14725,31027,...];
P(x)^(5^2): [1,25,(350),3575,29575,209405,1312675,7452225,...];
P(x)^(5^3): [1,125,8000,(349125),11676000,318906400,...];
P(x)^(5^4): [1,625,196250,41276875,(6541895625),833314453875,...];
P(x)^(5^5): [1,3125,4887500,5100915625,3996555181250,(2507423437503750),..];
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A000041, A158102, A158103, A158104, A158112, A158113, A158114, A158115.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[1/QPochhammer[q]^(5^n), {q, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Nov 24 2015 *)
  • PARI
    {a(n)=polcoeff(1/eta(x+x*O(x^n))^(5^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(5^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(5^m*x)^k/(1-(5^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(5^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (5^n*x)^k/(1 - (5^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] P(x)^(5^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).

A158114 a(n) = [x^n] eta(x)^(4^n).

Original entry on oeis.org

1, -4, 104, -37632, 166534720, -9109541173248, 6487005386806124544, -62637995710787181892993024, 8428730138560436521519921925857280, -16103390694987849639716307556519680725483520
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 - 4*x + 104*x^2 - 37632*x^3 + 166534720*x^4 +...
A(x) = 1 + log(eta(4*x)) + log(eta(16*x))^2/2! + log(eta(64*x))^3/3! +...
...
Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
then a(n) is the coefficient of x^n in eta(x)^(4^n):
eta(x)^(4^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..];
eta(x)^(4^1): [1,(-4),2,8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...];
eta(x)^(4^2): [1,-16,(104),-320,260,1248,-3712,1664,6890,...];
eta(x)^(4^3): [1,-64,1952,(-37632),512400,-5207936,40618368,...];
eta(x)^(4^4): [1,-256,32384,-2698240,(166534720),-8118668800,...];
eta(x)^(4^5): [1,-1024,522752,-177385472,45010254080,(-9109541173248), ...];
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A010815, A158112, A158113, A158115, A158102, A158103, A158104, A158105.

Programs

  • PARI
    {a(n)=polcoeff(eta(x+x*O(x^n))^(4^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,log(eta(4^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(4^m*x)^k/(1-(4^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} log( eta(4^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (4^n*x)^k/(1 - (4^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] Product_{k>=1} (1-x^k)^(4^n).

A158115 a(n) = [x^n] eta(x)^(5^n).

Original entry on oeis.org

1, -5, 275, -302250, 6175682500, -2459739648441250, 20152832471795703093750, -3521676074865217676579415546875, 13442076416943428772681311252971648437500
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2009

Keywords

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Examples

			G.f.: A(x) = 1 - 5*x + 275*x^2 - 302250*x^3 + 6175682500*x^4 +...
A(x) = 1 + log(eta(5*x)) + log(eta(25*x))^2/2! + log(eta(125*x))^3/3! +...
...
Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
then a(n) is the coefficient of x^n in eta(x)^(5^n):
eta(x)^(5^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,..];
eta(x)^(5^1): [1,(-5),5,10,-15,-6,-5,25,15,-20,9,-45,-5,25,...];
eta(x)^(5^2): [1,-25,(275),-1700,6050,-9405,-15550,107525,...];
eta(x)^(5^3): [1,-125,7625,(-302250),8745875,-196718900,...];
eta(x)^(5^4): [1,-625,194375,-40105000,(6175682500),...];
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A010815, A158112, A158113, A158114, A158102, A158103, A158104, A158105.

Programs

  • PARI
    {a(n)=polcoeff(eta(x+x*O(x^n))^(5^n), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,log(eta(5^m*x+x*O(x^n)))^m/m!), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(5^m*x)^k/(1-(5^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} log( eta(5^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (5^n*x)^k/(1 - (5^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] Product_{k>=1} (1-x^k)^(5^n).

A166953 Number of ways of writing n as the sum of 3^n squares.

Original entry on oeis.org

1, 6, 144, 23400, 26620002, 216778910040, 13069351570163616, 6019308484501930839936, 21708290476794620365667887680, 624502420526473667139055032092300382
Offset: 0

Views

Author

Paul D. Hanna, Oct 26 2009

Keywords

Examples

			G.f.: A(x) = 1 + 6*x + 144*x^2 + 23400*x^3 + 26620002*x^4 +...
Let F(x) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2),
then A(x) = 1 + log(F(3*x)) + log(F(9*x))^2/2! + log(F(27*x))^3/3! + log(F(81*x))^4/4! + ...
Illustrate a(n) = [x^n] F(x)^(3^n) by forming a table of
coefficients in powers F(x)^(3^n), which begin:
F^(3^0): [(1), 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, ...];
F^(3^1): [1, (6), 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, ...];
F^(3^2): [1, 18, (144), 672, 2034, 4320, 7392, 12672, 22608, ...];
F^(3^3): [1, 54, 1404, (23400), 280854, 2586168, 19014840, ...];
F^(3^4): [1, 162, 12960, 682560, (26620002), 819916992, ...];
F^(3^5): [1, 486, 117612, 18896328, 2267559846, (216778910040), ...]; ...
and noting that the coefficients along the diagonal (in parenthesis)
form the initial terms of this sequence.

Crossrefs

Cf. A000122, A005875, A008452, variants: A166947, A158113.

Programs

  • PARI
    {a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(3^n),n)}
  • PARI
    {a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n)); polcoeff(sum(k=0,n,log(subst(THETA3,x,3^k*x))^k/k!),n)}

Formula

a(n) equals the coefficient of x^n in the (3^n)-th power of Jacobi theta_3(x).
G.f.: A(x) = Sum_{n>=0} log( theta_3(3^n*x) )^n/n! where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
Showing 1-8 of 8 results.

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