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 1-8 of 8 results.

A358953 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(3*n) * (x^n - 2*A(x))^(4*n+1).

Original entry on oeis.org

1, 3, 21, 159, 1369, 12131, 111489, 1042310, 9878188, 94345595, 905236045, 8698907855, 83509981377, 798911473287, 7596665295846, 71585365842419, 666055801137389, 6089025714101416, 54304588402962717, 467144137463862047, 3798557443794080777, 27983895459969702990
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 159*x^3 + 1369*x^4 + 12131*x^5 + 111489*x^6 + 1042310*x^7 + 9878188*x^8 + 94345595*x^9 + 905236045*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(3*n) * (x^n - 2*Ser(A))^(4*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

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

A358954 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(4*n) * (x^n - 2*A(x))^(5*n+1).

Original entry on oeis.org

1, 4, 36, 384, 4568, 57920, 768760, 10543120, 148247390, 2125715618, 30965114225, 456956616284, 6817011617601, 102640570550600, 1557716916728198, 23804070258610024, 365964582592739540, 5656501536118793076, 87846324474413129008, 1370097609728212588634, 21451062781643458337802
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 4*x + 36*x^2 + 384*x^3 + 4568*x^4 + 57920*x^5 + 768760*x^6 + 10543120*x^7 + 148247390*x^8 + 2125715618*x^9 + 30965114225*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(4*n) * (x^n - 2*Ser(A))^(5*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

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

A358955 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(5*n) * (x^n - 2*A(x))^(6*n+1).

Original entry on oeis.org

1, 5, 55, 715, 10285, 157577, 2521339, 41635879, 704264465, 12139738505, 212475103777, 3765897874074, 67454279084444, 1219122315546851, 22204489538545069, 407150017658467685, 7509869807043464691, 139245172845883281403, 2593887890033997265241, 48521833007161546858193
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 5*x + 55*x^2 + 715*x^3 + 10285*x^4 + 157577*x^5 + 2521339*x^6 + 41635879*x^7 + 704264465*x^8 + 12139738505*x^9 + 212475103777*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(5*n) * (x^n - 2*Ser(A))^(6*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

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

A358956 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(6*n) * (x^n - 2*A(x))^(7*n+1).

Original entry on oeis.org

1, 6, 78, 1196, 20280, 366288, 6908744, 134492752, 2681961056, 54504790720, 1124768357872, 23505633975616, 496452504891320, 10580216111991080, 227237269499825185, 4913552644294206262, 106877300690757456293, 2336971970184440328572, 51339570414117180476064
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 6*x + 78*x^2 + 1196*x^3 + 20280*x^4 + 366288*x^5 + 6908744*x^6 + 134492752*x^7 + 2681961056*x^8 + 54504790720*x^9 + 1124768357872*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(6*n) * (x^n - 2*Ser(A))^(7*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(6*n) * (x^n - 2*A(x))^(7*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(7*n*(n-1)) / (1 - 2*A(x)*x^n)^(7*n-1).

A358957 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(7*n) * (x^n - 2*A(x))^(8*n+1).

Original entry on oeis.org

1, 7, 105, 1855, 36225, 753319, 16356809, 366518975, 8412321985, 196761671175, 4672976571753, 112386313863327, 2731613284143345, 66992673654966087, 1655756220596437601, 41199365822954474670, 1031225066096367871764, 25947188077245338061147, 655925022779049206277461
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 7*x + 105*x^2 + 1855*x^3 + 36225*x^4 + 753319*x^5 + 16356809*x^6 + 366518975*x^7 + 8412321985*x^8 + 196761671175*x^9 + 4672976571753*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(7*n) * (x^n - 2*Ser(A))^(8*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(7*n) * (x^n - 2*A(x))^(8*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(8*n*(n-1)) / (1 - 2*A(x)*x^n)^(8*n-1).

A358958 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(8*n) * (x^n - 2*A(x))^(9*n+1).

Original entry on oeis.org

1, 8, 136, 2720, 60112, 1414400, 34744192, 880722944, 22866372480, 604987038208, 16252230833792, 442118711113216, 12154717695451712, 337169716435693120, 9425612400257630864, 265272780558100130464, 7510038750103097772890, 213729057394800722424678, 6110972702751703321123745
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 8*x + 136*x^2 + 2720*x^3 + 60112*x^4 + 1414400*x^5 + 34744192*x^6 + 880722944*x^7 + 22866372480*x^8 + 604987038208*x^9 + 16252230833792*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(8*n) * (x^n - 2*Ser(A))^(9*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(8*n) * (x^n - 2*A(x))^(9*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(9*n*(n-1)) / (1 - 2*A(x)*x^n)^(9*n-1).

A358959 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(9*n) * (x^n - 2*A(x))^(10*n+1).

Original entry on oeis.org

1, 9, 171, 3819, 94221, 2474541, 67842255, 1919233719, 55608288057, 1641837803793, 49218744365683, 1494112796918051, 45836491198618821, 1418839143493455861, 44259772786526485527, 1389967891240928450511, 43910122539568806384513, 1394423517592589134138485
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 9*x + 171*x^2 + 3819*x^3 + 94221*x^4 + 2474541*x^5 + 67842255*x^6 + 1919233719*x^7 + 55608288057*x^8 + 1641837803793*x^9 + 49218744365683*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^(9*n) * (x^n - 2*Ser(A))^(10*n+1) ), #A-1)/2);A[n+1]}
    for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(9*n) * (x^n - 2*A(x))^(10*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(10*n*(n-1)) / (1 - 2*A(x)*x^n)^(10*n-1).

A361766 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n/A(-x))^(n+2).

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 57, 123, 280, 666, 1614, 3955, 9733, 23949, 58967, 145844, 363137, 910339, 2295192, 5811070, 14754567, 37542078, 95715596, 244567665, 626388406, 1608131393, 4137707994, 10667045757, 27546269363, 71241831762, 184508259405, 478501423792
Offset: 0

Views

Author

Paul D. Hanna, Mar 26 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n) * (y - x^n)^(n+2), which holds for all y as a formal power series in x. - Paul D. Hanna, Jan 19 2025

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 57*x^6 + 123*x^7 + 280*x^8 + 666*x^9 + 1614*x^10 + 3955*x^11 + 9733*x^12 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
    A[#A] = -polcoeff( sum(m=-#A,#A, (-x)^m * (1 - (-x)^m/Ser(A))^(m+2) ), #A-3));A[n+1]}
    for(n=0,35,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) - (-x)^n)^(n+2) / A(x)^n.
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) * A(x)^n / (1 - (-x)^n*A(x))^(n-2).
From Paul D. Hanna, Jan 19 2025: (Start)
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (1 - (-x)^n)^n * A(x)^n.
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) / ((1 - (-x)^n)^n * A(x)^n).
(End)
a(n) ~ c * d^n / n^(3/2), where d = 2.71312501383... and c = 3.43853109... - Vaclav Kotesovec, Mar 31 2023
Showing 1-8 of 8 results.