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.

A357160 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 2, 8, 24, 88, 313, 1187, 4549, 17898, 71324, 288365, 1177729, 4856051, 20178061, 84427850, 355375253, 1503849591, 6394015744, 27301536104, 117020066991, 503313598572, 2171633107742, 9396938664272, 40769489510945, 177313714453588, 772906669281227, 3376119803594888
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A356783.
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 + 2*x^2 + 8*x^3 + 24*x^4 + 88*x^5 + 313*x^6 + 1187*x^7 + 4549*x^8 + 17898*x^9 + 71324*x^10 + ...
such that
1 = ... + 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)^3 = ... + 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. A356783, A357161, A357162, A357163, A357164, A357165.

Programs

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

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.

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.

A357202 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 2, 9, 35, 182, 921, 5062, 28234, 162330, 947773, 5622641, 33747694, 204676547, 1252083028, 7717376754, 47878314072, 298749048454, 1873637869199, 11804288518884, 74673607921030, 474128308291896, 3020493580980524, 19301224674496592, 123681469340775568
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357152 and A357162.
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 + 2*x^2 + 9*x^3 + 35*x^4 + 182*x^5 + 921*x^6 + 5062*x^7 + 28234*x^8 + 162330*x^9 + 947773*x^10 + 5622641*x^11 + 33747694*x^12 + ...
such that
A(x)^2 = ... + 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)^5 = ... + 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. A357152, A357162, A357200, A357201, A357203, A357204, A357205.

Programs

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