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.

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A357404 Coefficients in the power series A(x) such that: 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 4, 32, 324, 3632, 43640, 549472, 7154952, 95563392, 1301943972, 18022506736, 252768034908, 3584103003152, 51294399688504, 739984677348512, 10749373940462452, 157101410692820448, 2308378616597302488, 34080671255517914992, 505321131709023383016, 7521442675843527317728
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
a(n) = Sum_{k=0..n} A357400(n,k) * 4^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 4*x + 32*x^2 + 324*x^3 + 3632*x^4 + 43640*x^5 + 549472*x^6 + 7154952*x^7 + 95563392*x^8 + 1301943972*x^9 + 18022506736*x^10 + ...
such that
4 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-4*A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357400, A356783, A357402, A357403, A357405.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(4 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) 4*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -4*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -4*A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

A357405 Coefficients in the power series A(x) such that: 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 5, 50, 630, 8825, 132490, 2084115, 33903705, 565697930, 9627904690, 166493454330, 2917050253615, 51670197054515, 923774673549045, 16647699155752645, 302098954307654995, 5515438344643031325, 101237254225602624790, 1867129260849076888865, 34583287418814030368150
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
a(n) = Sum_{k=0..n} A357400(n,k) * 5^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 8825*x^4 + 132490*x^5 + 2084115*x^6 + 33903705*x^7 + 565697930*x^8 + 9627904690*x^9 + 166493454330*x^10 + ...
such that
5 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-5*A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357400, A356783, A357402, A357403, A357404.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(5 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) 5*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -5*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -5*A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

A380557 G.f. satisfies A(x) such that: -1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 2, 10, 35, 146, 589, 2521, 10880, 48130, 215490, 978131, 4483493, 20740309, 96667511, 453596099, 2140879339, 10157274086, 48414142443, 231726319442, 1113290775079, 5366873616498, 25952658569610, 125856499026093, 611930422986515, 2982444057333882, 14568259180879990, 71307949455547118
Offset: 0

Views

Author

Paul D. Hanna, Feb 03 2025

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).
Note that formulas (5) and (6) are identities, and hold true for all A(x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 35*x^4 + 146*x^5 + 589*x^6 + 2521*x^7 + 10880*x^8 + 48130*x^9 + 215490*x^10 + ...
SPECIFIC VALUES.
A(t) = 15/8 at t = 0.19344501240894726710748422307503613491843235983978...
  where -1 = Sum_{n=-oo..+oo} (-1)^n * t^(2*n+1) * (1 + t^n)^(n+1) * (15/8)^n.
A(t) = 13/7 at t = 0.19341948378562934846535490742010025888491204467175...
A(t) = 11/6 at t = 0.19333502076470314454576717568898264806286912451280...
A(t) = 9/5 at t = 0.193110973645115287451084966528093291445869685605026...
A(t) = 7/4 at t = 0.192511645242345015361112270688385360547743653185979...
A(t) = 5/3 at t = 0.190649553303712199475798636706794101316873079578727...
A(t) = 3/2 at t = 0.182089586086018008207410926078691444238166561231377...
A(t) = 4/3 at t = 0.161675866655112310035152981730415472582224089685922...
A(t) = 5/4 at t = 0.143001255997678107192529149738503806026990657450325...
A(t) = 6/5 at t = 0.127286533527611786785145642678412658294861536180040...
A(t) = 7/6 at t = 0.114247661034580905508079370420172649525885310579285...
A(1/6) = 1.3631240552377275579566206545056633589020532732074...
  where -1 = Sum_{n=-oo..+oo} (-1)^n * (1/6)^(2*n+1) * (1 + (1/6)^n)^(n+1) * A(1/6)^n.
A(1/7) = 1.2494768685846922246570903862376666502561254090745...
A(1/8) = 1.1937090558071312140144246419584402049019509862828...
A(1/9) = 1.1594652229281839152092617957390758203214375656645...
A(1/10) = 1.135997746902180909378823046338236460433675615420...

Links

Crossrefs

Cf. A356783.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(1 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
(2) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.
(3) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).
(4) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).

A382322 G.f. A(x) satisfies -2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 2, 8, 50, 308, 2044, 14072, 100172, 730328, 5428498, 40978780, 313322910, 2421454020, 18884988540, 148443853936, 1174814738082, 9353539487160, 74865615299260, 602057472027484, 4862177553583604, 39416710563473400, 320650120976612168, 2616673301770051376, 21414973020645504142
Offset: 0

Views

Author

Paul D. Hanna, Mar 21 2025

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).
Formulas (5) and (6) hold true for all nonzero A(x).

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 50*x^3 + 308*x^4 + 2044*x^5 + 14072*x^6 + 100172*x^7 + 730328*x^8 + 5428498*x^9 + 40978780*x^10 + ...

Links

Crossrefs

Cf. A356783, A380557, A382323.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoef(2 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.
(1) -2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
(2) 2*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.
(3) -2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).
(4) 2*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).

A382323 G.f. A(x) satisfies -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 3, 18, 150, 1323, 12486, 123069, 1253595, 13089576, 139367370, 1507353966, 16515098985, 182913374493, 2044565139303, 23035036108755, 261312501113193, 2982280058702499, 34217698991867058, 394470188685557271, 4566935001939261414, 53076293916648500439, 618991948535588040078
Offset: 0

Views

Author

Paul D. Hanna, Mar 21 2025

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).
Formulas (5) and (6) hold true for all nonzero A(x).

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 150*x^3 + 1323*x^4 + 12486*x^5 + 123069*x^6 + 1253595*x^7 + 13089576*x^8 + 139367370*x^9 + ...

Links

Crossrefs

Cf. A356783, A380557, A382322.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoef(3 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.
(1) -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
(2) 3*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.
(3) -3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).
(4) 3*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).
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