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 1-10 of 15 results. Next

A358961 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).

Original entry on oeis.org

1, 3, 7, 33, 163, 858, 4708, 26662, 154699, 914885, 5494719, 33423598, 205493244, 1274928510, 7972042450, 50188844583, 317861388939, 2023777490895, 12945901676736, 83163975425669, 536279878717858, 3470134399230086, 22525040920670333, 146633283078321531
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 3*x + 7*x^2 + 33*x^3 + 163*x^4 + 858*x^5 + 4708*x^6 + 26662*x^7 + 154699*x^8 + 914885*x^9 + 5494719*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-3))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x)^(-1) + x*(A - x^3)^0 + x^2*(A - x^5) + x^3*(A - x^7)^2 + x^4*(A - x^9)^3 + ... + x^n * (A - x^(2*n+1))^(n-1) + ...
also,
A(x) = ... + x^24/(1 - x^(-7)*A)^(-2) - x^12/(1 - x^(-5)*A)^(-1) + x^4 - 1/(1 - x^(-1)*A) + 1/(1 - x*A)^2 - x^4/(1 - x^3*A)^3 + x^12/(1 - x^5*A)^4 - x^24/(1 - x^7*A)^5 + ... + (-1)^(n+1)*x^(2*n*(n-1))/(1 - x^(2*n-1)*A(x))^(n+1) + ...

Links

Crossrefs

Cf. A357227, A358962, A358963, A358964, A358965, A358937.

Programs

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

Formula

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

A363312 Expansion of g.f. A(x) satisfying 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 3.

Original entry on oeis.org

3, 8, 68, 656, 6924, 77816, 912504, 11043616, 136909712, 1729812880, 22193496988, 288368706416, 3786876943856, 50180784019384, 670150485880336, 9010466250798080, 121871951481594296, 1657086342551799752, 22637216782139196588, 310547100988853539728
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 2^2) for n > 0.

Examples

			G.f.: A(x) = 3 + 8*x + 68*x^2 + 656*x^3 + 6924*x^4 + 77816*x^5 + 912504*x^6 + 11043616*x^7 + 136909712*x^8 + 1729812880*x^9 + ...

Links

Crossrefs

Cf. A357227, A363141, A363313, A363314, A363315.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).

A363313 Expansion of g.f. A(x) satisfying 1/3 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 4.

Original entry on oeis.org

4, 18, 216, 3006, 46062, 752058, 12824370, 225765756, 4072115322, 74865020256, 1397774141280, 26431211243142, 505157673609054, 9742590254518956, 189370217827381284, 3705934209907310622, 72957899444047650828, 1443901345003970392266, 28710711213830156663136
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 3^2) for n > 0.

Examples

			G.f.: A(x) = 4 + 18*x + 216*x^2 + 3006*x^3 + 46062*x^4 + 752058*x^5 + 12824370*x^6 + 225765756*x^7 + 4072115322*x^8 + 74865020256*x^9 + ...

Links

Crossrefs

Cf. A357227, A363141, A363312, A363314, A363315.

Programs

  • PARI
    {a(n) = my(A=[4]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-3  + 3^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1/3 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/3 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/3 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/3 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).

A363314 Expansion of g.f. A(x) satisfying 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 5.

Original entry on oeis.org

5, 32, 496, 9024, 181296, 3882848, 86887712, 2007577472, 47530180736, 1147071160768, 28114384217104, 697913487791552, 17511114852998912, 443374443981736160, 11314170816869911232, 290688529521060711424, 7513202655833624201472, 195216134898681278515232
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 4^2) for n > 0.

Examples

			G.f.: A(x) =  5 + 32*x + 496*x^2 + 9024*x^3 + 181296*x^4 + 3882848*x^5 + 86887712*x^6 + 2007577472*x^7 + 47530180736*x^8 + ...

Links

Crossrefs

Cf. A357227, A363141, A363312, A363313, A363315.

Programs

  • PARI
    {a(n) = my(A=[5]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-4  + 4^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/4 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).

A363315 Expansion of g.f. A(x) satisfying 1/5 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 6.

Original entry on oeis.org

6, 50, 950, 21350, 530700, 14067650, 389701050, 11147799700, 326779719500, 9764739197800, 296342706620800, 9108989853295550, 283002934668287000, 8872796279035164100, 280368062326854982450, 8919740526808334086550, 285476263708658548421000, 9185078302539674382641450
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 5^2) for n > 0.

Examples

			G.f.: A(x) = 6 + 50*x + 950*x^2 + 21350*x^3 + 530700*x^4 + 14067650*x^5 + 389701050*x^6 + 11147799700*x^7 + 326779719500*x^8 + ...

Links

Crossrefs

Cf. A357227, A363141, A363312, A363313, A363314.

Programs

  • PARI
    {a(n) = my(A=[6]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-5  + 5^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

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

A361772 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(2*n-1).

Original entry on oeis.org

1, 1, 8, 61, 600, 6072, 65804, 733435, 8415694, 98529785, 1173278329, 14162417506, 172914841649, 2131621288494, 26495818020038, 331706510158239, 4178800564364333, 52935845003315662, 673878770026778330, 8616336680850069832, 110606714769468383785, 1424933340070339610543
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 61*x^3 + 600*x^4 + 6072*x^5 + 65804*x^6 + 733435*x^7 + 8415694*x^8 + 98529785*x^9 + 1173278329*x^10 + ...

Links

Crossrefs

Cf. A361771, A361773, A361774.
Cf. A363112, A355865, A357227, A359712, A357232.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(2*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n^2) / (1 - 2*A(x)*(-x)^n)^(2*n+1).

A361773 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(3*n-1).

Original entry on oeis.org

1, 2, 34, 677, 15660, 393790, 10433402, 286990626, 8117763488, 234635708480, 6899771599141, 205768408153474, 6208628685564955, 189188990142419693, 5813805339043713267, 179968235623379467274, 5606627898452185950618, 175650401043239524832783, 5530500462355496324862920
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 34*x^2 + 677*x^3 + 15660*x^4 + 393790*x^5 + 10433402*x^6 + 286990626*x^7 + 8117763488*x^8 + 234635708480*x^9 + 6899771599141*x^10 + ...

Links

Crossrefs

Cf. A361771, A361772, A361774.
Cf. A363113, A355865, A357227, A359712, A357232.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(3*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n^2) / (1 - 2*A(x)*(-x)^n)^(3*n+1).

A363141 Expansion of g.f. A(x) satisfying 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 0, 2, 3, 11, 23, 76, 188, 575, 1587, 4732, 13714, 40993, 121787, 367100, 1107371, 3367412, 10267404, 31468401, 96734992, 298488537, 923587457, 2866241029, 8916951360, 27808418089, 86910042122, 272180834822, 854004007736, 2684311988984, 8451232727631
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^3 + 3*x^4 + 11*x^5 + 23*x^6 + 76*x^7 + 188*x^8 + 575*x^9 + 1587*x^10 + 4732*x^11 + 13714*x^12 + ...

Links

Crossrefs

Cf. A357227, A363312, A363313, A363314, A363315.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/x = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 3.3064656105288278... and c = 0.3845291573508... - Vaclav Kotesovec, Jun 09 2023

A361771 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(n-1).

Original entry on oeis.org

1, 1, 1, 7, 28, 89, 421, 1898, 7912, 36412, 169960, 779139, 3668210, 17486938, 83333003, 400956919, 1943928504, 9455346485, 46225027071, 227066384875, 1119123274755, 5534782142253, 27463607765186, 136652474592260, 681728348606011, 3409395265172439, 17088672210734316
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 28*x^4 + 89*x^5 + 421*x^6 + 1898*x^7 + 7912*x^8 + 36412*x^9 + 169960*x^10 + 779139*x^11 + 3668210*x^12 + ...

Links

Crossrefs

Cf. A361772, A361773, A361774.
Cf. A355865, A357227, A359712, A357232.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*A(x)*(-x)^n)^(n+1).

A361774 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(4*n-1).

Original entry on oeis.org

1, 4, 150, 7003, 380817, 22517717, 1405927141, 91215539609, 6089092570148, 415519886498886, 28855638743197866, 2032628861705203315, 144884697917577076857, 10430845410431559928714, 757390467820895322043476, 55401570124877193188443429, 4078685155312165112343519832
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 150*x^2 + 7003*x^3 + 380817*x^4 + 22517717*x^5 + 1405927141*x^6 + 91215539609*x^7 + 6089092570148*x^8 + ...

Crossrefs

Cf. A361771, A361772, A361773.
Cf. A363114, A355865, A357227, A359712, A357232.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(4*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n^2) / (1 - 2*A(x)*(-x)^n)^(4*n+1).
Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.