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 1-9 of 9 results.

A214690 Triangle, read by rows of n^2 terms, where row n equals the coefficients in the series reversion of the function G(y,n)-1 such that: y = Sum_{m>=1} 1/G(y,n)^(2*n*m) * Product_{k=1..m} (1 - 1/G(y,n)^(2*k-1)).

Original entry on oeis.org

1, 1, -2, -3, -1, 1, -4, -2, 22, 49, 49, 27, 8, 1, 1, -6, 3, 61, 15, -567, -1946, -3607, -4489, -4015, -2640, -1274, -441, -104, -15, -1, 1, -8, 12, 108, -218, -1938, -834, 27124, 136919, 393601, 809873, 1288950, 1646268, 1720788, 1487263, 1067345, 635682, 312646
Offset: 1

Views

Author

Paul D. Hanna, Jul 25 2012

Keywords

Examples

			Consider the family of power series G(x,n) that satisfy:
x = Sum_{m>=1} 1/G(x,n)^(2*n*m) * Product_{k=1..m} (1 - 1/G(x,n)^(2*k-1)).
Examples of sequences with g.f. G(x,n) are:
n=2: A214692 = [1, 1, 2, 11, 71, 515, 3997, 32488, 273009, ...];
n=3: A214693 = [1, 1, 4, 34, 338, 3691, 42623, 510949, 6289912, ...];
n=4: A214694 = [1, 1, 6, 69, 929, 13692, 213402, 3456450, ...];
n=5: A214695 = [1, 1, 8, 116, 1972, 36682, 722098, 14784834, ...]; ...
Observe that Series_Reversion(G(x,n) - 1) is given by the polynomials:
n=1: x;
n=2: x - 2*x^2 - 3*x^3 - x^4;
n=3: x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9;
n=4: x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 - 1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16; ...
This triangle of coefficients in the above polynomials begins:
[1];
[1, -2, -3, -1];
[1, -4, -2, 22, 49, 49, 27, 8, 1];
[1, -6, 3, 61, 15, -567, -1946, -3607, -4489, -4015, -2640, -1274, -441, -104, -15, -1];
[1, -8, 12, 108, -218, -1938, -834, 27124, 136919, 393601, 809873, 1288950, 1646268, 1720788, 1487263, 1067345, 635682, 312646, 125761, 40734, 10373, 2001, 275, 24, 1];
[1, -10, 25, 155, -750, -3562, 12824, 113082, 113375, -2035735, -14707914, -59955129, -179036484, -426054391, -841492130, -1412100002, -2043288274, -2574420276, -2842741390, -2762638817, -2368603455, -1793326192, -1198603784, -706071990, -365534676, -165596757, -65259715, -22195440, -6446730, -1576815, -318649, -51799, -6511, -594, -35, -1]; ...

Links

Crossrefs

Cf. A214691 (row sums), A214692, A214693, A214694, A214695, A214670 (variant).

Programs

  • PARI
    {T(n, k)=local(Axy=x*y); Axy=sum(m=1, n, -x^m*prod(j=1, m, (1-(1+y)^(2*j-1))/(1-x*(1+y)^(2*j-1))+x*O(x^n))); polcoeff(polcoeff(Axy, n, x), k, y)}
{for(n=1, 10, for(k=1, n^2, print1(T(n, k), ", ")); print(""))}
  • PARI
    {a(n, p)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(2*p*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
{for(n=1, 8, Tn=Vec(serreverse(sum(m=1, n^2, a(m, n)*x^m)+x*O(x^(n^2)))); for(k=1, n^2, print1(Tn[k], ", ")); print(""))}

Formula

G.f.: A(x,y) = Sum_{n>=1} -x^n * Product_{k=1..n} (1 - (1+y)^(2*k-1)) / (1 - x*(1+y)^(2*k-1)).
G.f. for row n is R(y,n) = Sum_{k=1..n^2} y^k*T(n,k) defined by:
A(x,y) = Sum_{n>=1} x^n * R(y,n) such that:
R(y,n) = Series_Reversion( G(y,n) - 1 ) where G(y,n) satisfies:
y = Sum_{m>=1} 1/G(y,n)^(2*n*m) * Product_{k=1..m} (1 - 1/G(y,n)^(2*k-1)), for n>=1.
Row polynomials R(y,n) satisfy:
(1) R(1,n) = -(-1)^n * A214691(n) for n>=1.
(2) R(-1,n) = 1 for n>=1.
(3) R'(-1,n) = 0 for n>1.

A214693 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 4, 34, 338, 3691, 42623, 510949, 6289912, 78972928, 1006665781, 12985611054, 169115724583, 2219614920740, 29318819296959, 389331204757856, 5192978617937181, 69522908878900079, 933674035184058960, 12571898958515379108, 169651868248129552194
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 338*x^4 + 3691*x^5 + 42623*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^3-1)/A(x)^16 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^27 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^40 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^55 +...

Crossrefs

Cf. A214690, A214692, A214694, A214695, A181997 (variant).

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9 +x^2*O(x^n)), n))}
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(6*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y = x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9, which is the g.f. of row 3 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+6)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

A214694 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(8*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 6, 69, 929, 13692, 213402, 3456450, 57585400, 980408857, 16982002433, 298322996205, 5302587890821, 95196447689434, 1723782813066284, 31447947375375315, 577509675356805547, 10667460556561578780, 198074286156460874227, 3695152948440645726312
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 69*x^3 + 929*x^4 + 13692*x^5 + 213402*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^9 + (A(x)-1)*(A(x)^3-1)/A(x)^20 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^33 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^48 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^65 +...

Crossrefs

Cf. A214690, A214692, A214693, A214695, A181998 (variant).

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 -
1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16 +x^2*O(x^n)), n))}
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(8*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y = x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 - 1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16, which is the g.f. of row 4 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+8)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

A214695 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(10*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 8, 116, 1972, 36682, 722098, 14784834, 311629580, 6716892893, 147372681787, 3280609461927, 73912217824094, 1682234535898788, 38621258859241912, 893358073179541313, 20800314016777824187, 487100732909778007223, 11465386711990265812207
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 116*x^3 + 1972*x^4 + 36682*x^5 + 722098*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^11 + (A(x)-1)*(A(x)^3-1)/A(x)^24 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^39 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^56 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^75 +...

Crossrefs

Cf. A214690, A214692, A214693, A214694, A209441 (variant).

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 8*x^2 + 12*x^3 + 108*x^4 - 218*x^5 - 1938*x^6 - 834*x^7 + 27124*x^8 + 136919*x^9 + 393601*x^10 +
809873*x^11 + 1288950*x^12 + 1646268*x^13 + 1720788*x^14 + 1487263*x^15 + 1067345*x^16 + 635682*x^17 + 312646*x^18 + 125761*x^19 + 40734*x^20 +
10373*x^21 + 2001*x^22 + 275*x^23 + 24*x^24 + x^25 +x^2*O(x^n)), n))}
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(10*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y is a 25-degree polynomial in x and is the g.f. of row 5 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+10)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

A249926 G.f. A(x) satisfies: 1+x = A(x)^2 + A(x)^4 - A(x)^5.

Original entry on oeis.org

1, 1, 3, 24, 229, 2449, 28035, 336100, 4165920, 52953884, 686517601, 9042628374, 120669757468, 1627932844657, 22166277534398, 304230231637560, 4204474770868230, 58458984141770754, 817176088436608178, 11477568712206346244, 161897000202383717334, 2292445680627209103645
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 229*x^4 + 2449*x^5 + 28035*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^4-1)/A(x)^17 + (A(x)-1)*(A(x)^4-1)*(A(x)^7-1)/A(x)^30 + (A(x)-1)*(A(x)^4-1)*(A(x)^7-1)*(A(x)^10-1)/A(x)^46 +
(A(x)-1)*(A(x)^4-1)*(A(x)^7-1)*(A(x)^10-1)*(A(x)^13-1)/A(x)^65 +...
Related expansions.
A(x)^2 = 1 + 2*x + 7*x^2 + 54*x^3 + 515*x^4 + 5500*x^5 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 136*x^3 + 1295*x^4 + 13816*x^5 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 190*x^3 + 1810*x^4 + 19316*x^5 +...
where 1+x = A(x)^2 + A(x)^4 - A(x)^5.

Crossrefs

Cf. A214692, A249927, A249928, A249929, A249930, A249931, A249932.

Programs

  • PARI
    /* From 1+x = A(x)^2 + A(x)^4 - A(x)^5: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^4-Ser(A)^5)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 3*x^2 - 6*x^3 - 4*x^4 - x^5 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 3*x^2 - 6*x^3 - 4*x^4 - x^5).
G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(3*k-2)).
G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(n*(3*n-1)/2+6*n) * Product_{k=1..n} (A(x)^(3*k-2) - 1).

A249934 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(3*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 1, 4, 19, 107, 671, 4600, 34218, 276415, 2439426, 23724674, 256361107, 3091554768, 41560590331, 618957882104, 10119509431084, 179887355572358, 3446915545155744, 70686674091569072, 1542478858735415921, 35650141769790146478, 869385516566240903091, 22299067147713040916568
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			A(x) = 1 + x + x^2 + 4*x^3 + 19*x^4 + 107*x^5 + 671*x^6 + 4600*x^7 + 34218*x^8 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^4 + (A(x)-1)*(A(x)^3-1)/A(x)^10 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^18 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^28 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^40 +...

Links

Crossrefs

Cf. A214692.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF^3,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* More efficient than PARI program, Vaclav Kotesovec, Nov 30 2014 *)
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(3*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+3)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
a(n) ~ exp(Pi^2/24) * 12^n * n^(n-1) / (sqrt(6) * exp(n) * Pi^(2*n-1)). - Vaclav Kotesovec, Dec 01 2014

A247480 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(5*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 3, 21, 172, 1557, 14937, 148870, 1523150, 15874211, 167584946, 1784250269, 19082848084, 204183773733, 2174724531143, 22887441573480, 235016048710027, 2294441979279215, 19936497820248076, 118333942636382173, -709004900481995789, -49850788347995316262
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2014

Keywords

Links

Crossrefs

Cf. A247482 (exponent=0), A247481 (exponent=1), A249934 (exponent=3), A214692 (exponent=4), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).
Cf. A214690, A214670.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF^5,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]

Formula

a(n) ~ c * 12^n * n^(n-2) / (exp(n) * Pi^(2*n)), where c = -sqrt(6) * Pi^3 * exp(5*Pi^2/24)/24 = -24.7341070998048267... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

A247481 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^n * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, -1, -1, -2, -14, -98, -822, -7948, -86590, -1046916, -13892842, -200653570, -3133064534, -52596852266, -944892417438, -18091297436248, -367841660947508, -7916992964642992, -179849204152350892, -4300928485463624458, -108013481381638292266
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2014

Keywords

Links

Crossrefs

Cf. A247482 (exponent=0), A249934 (exponent=3), A214692 (exponent=4), A247480 (exponent=5), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).
Cf. A214690, A214670.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]

Formula

a(n) ~ c * 12^n * n^n / (exp(n) * Pi^(2*n)), where c = -2*sqrt(6)/(Pi*exp(Pi^2/8)) = -0.45411558500969644... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

A247482 G.f. A(x) satisfies: x = Sum_{n>=1} Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, -2, 1, -3, -18, -124, -1174, -12150, -141536, -1816780, -25461723, -386593670, -6320496592, -110711177281, -2068814967831, -41089562943757, -864563028340432, -19214971769126974, -449887669808788433, -11069673481210168218, -285604488897863640237
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2014

Keywords

Links

Crossrefs

Cf. A247481 (exponent=1), A249934 (exponent=3), A214692 (exponent=4), A247480 (exponent=5), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).
Cf. A214690, A214670.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1)),{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A]=-polcoeff(sum(m=1, #A, prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2024, after Paul D. Hanna

Formula

a(n) ~ c * 12^n * n^(n+1/2) / (exp(n) * Pi^(2*n)), where c = -12 / (Pi^(3/2) * exp(5*Pi^2/24)) = -0.275723765924812729... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017
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