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 1-5 of 5 results.

A214669 a(n) = Sum_{k=1..n*(n+1)/2} k * A214670(n,k).

Original entry on oeis.org

1, -4, 36, -720, 29264, -2370368, 379341376, -119616350464, 74306729686272, -91021121260127232, 220137897341768508416, -1052620193410626774192128, 9963741205378506824566067200, -186912918064963001770860269879296, 6955904583765468148520677987319955456
Offset: 1

Views

Author

Paul D. Hanna, Jul 25 2012

Keywords

Crossrefs

Cf. A214670.

Programs

  • PARI
    {a(n)=local(A214670=x*y); A214670=sum(m=1, n, -x^m*prod(j=1, m, (1-(1+y)^j)/(1-x*(1+y)^j))+x*O(x^n)); polcoeff(subst(deriv(A214670,y),y,1), n, x)}
for(n=1, 25, print1(a(n),", "))

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.

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
Showing 1-5 of 5 results.

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