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 1-6 of 6 results.

A357161 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 3, 15, 71, 378, 2087, 12006, 70910, 428021, 2627731, 16358961, 103027423, 655236314, 4202210514, 27145925685, 176474644608, 1153679423108, 7579526316199, 50017854059557, 331390828183765, 2203548061830875, 14700363755114949, 98363233394747546
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357151.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 71*x^4 + 378*x^5 + 2087*x^6 + 12006*x^7 + 70910*x^8 + 428021*x^9 + 2627731*x^10 + ...
such that
A(x) = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^4 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357151, A357160, A357162, A357163, A357164, A357165.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A) - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#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) A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
(2) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
(3) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
(4) -A(x)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*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+2))^n.

A357162 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 4, 25, 162, 1160, 8731, 68364, 550707, 4535402, 38012170, 323168946, 2780229079, 24158457026, 211721412339, 1869239684558, 16609750957942, 148431230687412, 1333134683364035, 12027524448579488, 108951760865234373, 990555733683233240, 9035754580314840475
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357152.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 162*x^4 + 1160*x^5 + 8731*x^6 + 68364*x^7 + 550707*x^8 + 4535402*x^9 + 38012170*x^10 + ...
such that
A(x)^2 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^5 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357152, A357160, A357161, A357163, A357164, A357165.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A)^2 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#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) A(x)^2 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
(2) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
(3) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
(4) -A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*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+2))^n.

A357163 Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 38, 313, 2834, 27088, 269380, 2757797, 28872568, 307696566, 3326835855, 36403128996, 402370063992, 4485931975701, 50386112677647, 569624341701738, 6476615022560400, 74013180802610161, 849642206122063571, 9793310961240979983, 113297108937174512275
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357153.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 313*x^4 + 2834*x^5 + 27088*x^6 + 269380*x^7 + 2757797*x^8 + 28872568*x^9 + 307696566*x^10 + ...
such that
A(x)^3 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^6 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357153, A357160, A357161, A357162, A357164, A357165.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A)^3 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#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) A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
(2) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
(3) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
(4) -A(x)^6 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*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+2))^n.

A357164 Coefficients in the power series A(x) such that: A(x)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 6, 54, 540, 5925, 68753, 830267, 10324947, 131329213, 1700614790, 22344117822, 297132512955, 3991542148276, 54086668396101, 738390401404546, 10146440406910223, 140227571720595241, 1947883865390758591, 27181029295364007844, 380838895427784827916
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357154.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 540*x^4 + 5925*x^5 + 68753*x^6 + 830267*x^7 + 10324947*x^8 + 131329213*x^9 + 1700614790*x^10 + ...
such that
A(x)^4 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^7 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357154, A357160, A357161, A357162, A357163, A357165.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A)^4 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#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) A(x)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
(2) x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
(3) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
(4) -A(x)^7 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*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+2))^n.

A357165 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 7, 73, 859, 11083, 151369, 2151961, 31510682, 471993401, 7198166363, 111390268227, 1744706996712, 27606853938808, 440638645554932, 7086053148425023, 114700710907449375, 1867353232898846998, 30556409451787334011, 502291724376632138667, 8290605658533141188978
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357155.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 859*x^4 + 11083*x^5 + 151369*x^6 + 2151961*x^7 + 31510682*x^8 + 471993401*x^9 + 7198166363*x^10 + ...
such that
A(x)^5 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^8 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357155, A357160, A357161, A357162, A357163, A357164.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A)^5 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#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) A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
(2) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
(3) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
(4) -A(x)^8 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*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+2))^n.

A357200 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 0, 0, -7, -3, -17, 52, 51, 384, -227, -52, -6311, -2722, -18733, 79229, 67453, 620385, -619315, 85796, -13137380, -595833, -43282243, 205480697, 66895157, 1551910768, -2300631561, 1546386060, -36481481081, 15982958026, -135266506195, 652843485153
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A356783 and A357160.
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x - 7*x^4 - 3*x^5 - 17*x^6 + 52*x^7 + 51*x^8 + 384*x^9 - 227*x^10 - 52*x^11 - 6311*x^12 - 2722*x^13 - 18733*x^14 + ...
such that
1 = ... + x^(-2)*(1 - 1/x)^(-1)/A(x)^2 + x^(-1)/A(x) + (1 - x) + x*(1 - x^2)*A(x) + x^2*(1 - x^3)^3*A(x)^2 + x^3*(1 - x^4)^4*A(x)^3 + ... + x^n*(1 - x^(n+1))^(n+1)*A(x)^n + ...
also
-A(x)^3 = ... + x^(-2)*(A(x) - 1/x)^(-1)*A(x)^2 + x^(-1)*A(x) + (A(x) - x) + x*(A(x) - x^2)^2/A(x) + x^2*(A(x) - x^3)^3/A(x)^2 + x^3*(A(x) - x^4)^4/A(x)^3 + ... + x^n*(A(x) - x^(n+1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A356783, A357160, A357201, A357202, A357203, A357204, A357205.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(n=-#A-2, #A+2, x^n * (1 - x^(n+1) +x*O(x^#A))^(n+1) * Ser(A)^n  ), #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} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
(2) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^n)^n * A(x)^n ).
(3) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^n*A(x))^n.
(4) -A(x)^3 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1)*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)^n.
Showing 1-6 of 6 results.

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