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 1-4 of 4 results.

A317352 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).

Original entry on oeis.org

1, 1, 5, 42, 471, 6435, 102232, 1837630, 36719439, 805716679, 19239923577, 496514053880, 13769677836500, 408449335836132, 12906850662570996, 432942515731367894, 15367227978734187567, 575544844737119275935, 22685977410186834271463, 938867118118688412116554
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + 36719439*x^8 + 805716679*x^9 + ...
where A(x) = 1 + x*A(x)^3 + 2*x^2*A(x)^2*A'(x).
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*x*A(x)^2 ) / A(x) begins:
n=1: [1, 0, -5, -158, -7779, -563924, -56177105, -7318104450, ...];
n=2: [1, 1, 0, -94, -5968, -473688, -49352768, -6601523360, ...];
n=3: [1, 2, 7, 0, -3435, -354282, -40709709, -5723430444, ...];
n=4: [1, 3, 16, 130, 0, -199016, -29893568, -4657391616, ...];
n=5: [1, 4, 27, 302, 4541, 0, -16486865, -3372747590, ...];
n=6: [1, 5, 40, 522, 10416, 251976, 0, -1833979680, ...];
n=7: [1, 6, 55, 796, 17877, 567562, 20138467, 0, ...];
n=8: [1, 7, 72, 1130, 27200, 958968, 44592256, 2176638976, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A088716, A208961, A317353, A317354, A352236.

Programs

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

Formula

O.g.f. A(x) satisfies:
(1) A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).
(2) [x^n] exp( n * x*A(x)^2 ) / A(x) = 0 for n>0.
(3.a) [x^n] exp(-n * x*A(x)^2) * (2 - 1/A(x)) = 0 for n >= 1.
(3.b) [x^n] exp(-n^2 * x*A(x)^2) * (n + 1 - n/A(x)) = 0 for n >= 1.
(3.c) [x^n] exp(-n^(p+1) * x*A(x)^2) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.188286926603706833845600622... - Vaclav Kotesovec, Aug 05 2018

A352235 G.f. A(x) satisfies: A(x) = 1 + x*A(x) / (A(x) - 3*x*A'(x)).

Original entry on oeis.org

1, 1, 3, 24, 309, 5262, 108894, 2618718, 71246145, 2154788970, 71563126710, 2586270267600, 100995812044266, 4237522832234832, 190126298040192912, 9085093650185205498, 460711407231295513689, 24715373661154672634058, 1398648334415007990887454
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 309*x^4 + 5262*x^5 + 108894*x^6 + 2618718*x^7 + 71246145*x^8 + ...
such that A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+2) begins:
n=0: [1,  2,   7,   54,   675,  11286,   230742, ...];
n=1: [1,  5,  25,  190,  2210,  34981,   688635, ...];
n=2: [1,  8,  52,  416,  4642,  69872,  1322848, ...];
n=3: [1, 11,  88,  759,  8349, 120549,  2195886, ...];
n=4: [1, 14, 133, 1246, 13790, 193060,  3391017, ...];
n=5: [1, 17, 187, 1904, 21505, 295154,  5017618, ...];
n=6: [1, 20, 250, 2760, 32115, 436524,  7217250, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2), n >= 1,
as illustrated by
[x^1] A(x)^2 = 2 = [x^0] 2*A(x)^2 = 2*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^8 = 416 = [x^2] 8*A(x)^8 = 8*52;
[x^4] A(x)^11 = 8349 = [x^3] 11*A(x)^11 = 11*759;
[x^5] A(x)^14 = 193060 = [x^4] 14*A(x)^14 = 14*13790;
[x^6] A(x)^17 = 5017618 = [x^5] 17*A(x)^17 = 17*295154; ...

Links

Crossrefs

Cf. A088715, A286797, A317352, A352236, A352237, A352238.

Programs

  • PARI
    /* Using A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))
  • PARI
    /* Using [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2)*A(x)^(3*n+2) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A-2)+2) - Ser(A)^(3*(#A-2)+2)/(3*(#A-2)+2)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2) for n >= 1.
(2) A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
(3) A'(x) = A(x) * (1 + x/(1 - A(x))) / (3*x).
(4) A(x) = exp( Integral (1 + x/(1 - A(x))) / (3*x) dx ).
a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.09232038797888963484135336... - Vaclav Kotesovec, Nov 16 2023

A352237 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 3*x*A'(x)).

Original entry on oeis.org

1, 1, 4, 37, 532, 9994, 226252, 5910445, 173581060, 5634589906, 199792389160, 7671942375898, 316936631324368, 14011781050744984, 660054967923455212, 33008607551445324157, 1746771084107236755604, 97536010045722766992778, 5731874036042145864368824
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 37*x^3 + 532*x^4 + 9994*x^5 + 226252*x^6 + 5910445*x^7 + 173581060*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+1) begins:
n=0: [1,  1,   4,   37,   532,   9994,   226252, ...];
n=1: [1,  4,  22,  200,  2717,  48788,  1069122, ...];
n=2: [1,  7,  49,  462,  6069, 104664,  2219784, ...];
n=3: [1, 10,  85,  850, 11020, 183832,  3777355, ...];
n=4: [1, 13, 130, 1391, 18083, 294203,  5869734, ...];
n=5: [1, 16, 184, 2112, 27852, 445632,  8659920, ...];
n=6: [1, 19, 247, 3040, 41002, 650161, 12353059, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1), n >= 1,
as illustrated by
[x^1] A(x)^4 = 4 = [x^0] 4*A(x)^4 = 4*1;
[x^2] A(x)^7 = 49 = [x^1] 7*A(x)^7 = 7*7;
[x^3] A(x)^10 = 850 = [x^2] 10*A(x)^10 = 10*85;
[x^4] A(x)^13 = 18083 = [x^3] 13*A(x)^13 = 13*1391;
[x^5] A(x)^16 = 445632 = [x^4] 16*A(x)^16 = 16*27852;
[x^6] A(x)^19 = 12353059 = [x^5] 19*A(x)^19 = 19*650161; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^3) = 1 + x + 7*x^2 + 85*x^3 + 1391*x^4 + 27852*x^5 + 650161*x^6 + 17204220*x^7 + ...
where B(x)^3 = (1/x) * Series_Reversion( x/A(x)^3 ).

Links

Crossrefs

Cf. A088715, A286797, A317352, A352235, A352236, A352238.

Programs

  • PARI
    /* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))
  • PARI
    /* Using [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1)*A(x)^(3*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A)-2) - Ser(A)^(3*(#A)-2)/(3*(#A)-2)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (3*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(3*x) dx ).
a(n) ~ c * 3^n * n! * n^(4/3), where c = 0.0543186200722307001992331... - Vaclav Kotesovec, Nov 16 2023

A352238 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 4*x*A'(x)).

Original entry on oeis.org

1, 1, 5, 61, 1161, 28857, 864141, 29861749, 1160382737, 49854838897, 2340623599381, 119051103325613, 6516915195123097, 381912592990453545, 23856225840952434333, 1582482450123627473637, 111113139625779846025761, 8234335766045466358238433
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1161*x^4 + 28857*x^5 + 864141*x^6 + 29861749*x^7 + 1160382737*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 4*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(4*n+1) begins:
n=0: [1,  1,   5,   61,   1161,   28857,   864141, ...];
n=1: [1,  5,  35,  415,   7430,  176286,  5107530, ...];
n=2: [1,  9,  81,  993,  17127,  389583, 10916559, ...];
n=3: [1, 13, 143, 1859,  31564,  693212, 18802212, ...];
n=4: [1, 17, 221, 3077,  52309, 1118549, 29427153, ...];
n=5: [1, 21, 315, 4711,  81186, 1704906, 43640030, ...];
n=6: [1, 25, 425, 6825, 120275, 2500555, 62513875, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1) * A(x)^(4*n+1), n >= 1,
as illustrated by
[x^1] A(x)^5 = 5 = [x^0] 5*A(x)^5 = 5*1;
[x^2] A(x)^9 = 81 = [x^1] 9*A(x)^9 = 9*9;
[x^3] A(x)^13 = 1859 = [x^2] 13*A(x)^13 = 13*143;
[x^4] A(x)^17 = 52309 = [x^3] 17*A(x)^17 = 17*3077;
[x^5] A(x)^21 = 1704906 = [x^4] 21*A(x)^21 = 21*81186;
[x^6] A(x)^25 = 62513875 = [x^5] 25*A(x)^25 = 25*2500555; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^4) = 1 + x + 9*x^2 + 143*x^3 + 3077*x^4 + 81186*x^5 + 2500555*x^6 + 87388600*x^7 + ...
where B(x)^4 = (1/x) * Series_Reversion( x/A(x)^4 ).

Links

Crossrefs

Cf. A088715, A286797, A317352, A352235, A352236, A352237.

Programs

  • PARI
    /* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 4*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))
  • PARI
    /* Using [x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1)*A(x)^(4*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(4*(#A)-3) - Ser(A)^(4*(#A)-3)/(4*(#A)-3)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1) * A(x)^(4*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 4*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (4*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(4*x) dx ).
a(n) ~ c * 4^n * n! * n^(5/4), where c = 0.0440035900116077498469559... - Vaclav Kotesovec, Nov 16 2023
Showing 1-4 of 4 results.

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