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 1-5 of 5 results.

A352236 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 2*x*A'(x)).

Original entry on oeis.org

1, 1, 3, 19, 185, 2353, 36075, 638115, 12683761, 278485217, 6674259667, 173097575603, 4826128088489, 143896870347793, 4568544366818747, 153883892657000259, 5481761893234193889, 205939077652874352577, 8138639816942009694627, 337568614331296733526867
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 185*x^4 + 2353*x^5 + 36075*x^6 + 638115*x^7 + 12683761*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(2*n+1) begins:
n=0: [1,  1,   3,   19,   185,   2353,   36075, ...];
n=1: [1,  3,  12,   76,   705,   8595,  127680, ...];
n=2: [1,  5,  25,  165,  1490,  17506,  252050, ...];
n=3: [1,  7,  42,  294,  2632,  30016,  419454, ...];
n=4: [1,  9,  63,  471,  4239,  47295,  643017, ...];
n=5: [1, 11,  88,  704,  6435,  70785,  939312, ...];
n=6: [1, 13, 117, 1001,  9360, 102232, 1329016, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1), n >= 1,
as illustrated by
[x^1] A(x)^3 = 3 = [x^0] 3*A(x)^3 = 3*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^7 = 294 = [x^2] 7*A(x)^7 = 7*42;
[x^4] A(x)^9 = 4239 = [x^3] 9*A(x)^9 = 9*471;
[x^5] A(x)^11 = 70785 = [x^4] 11*A(x)^11 = 11*6435;
[x^6] A(x)^13 = 1329016 = [x^5] 13*A(x)^13 = 13*102232; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^2) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + ... + A317352(n)*x^n + ...
where B(x)^2 = (1/x) * Series_Reversion( x/A(x)^2 ).

Links

Crossrefs

Cf. A088715, A286797, A317352, A352235, A352237, A352238.

Programs

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

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (2*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(2*x) dx ).
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.06926688933886004638602492... - Vaclav Kotesovec, Nov 16 2023

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

A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 27, 40, 14, 1, 248, 419, 200, 30, 1, 2830, 5308, 3124, 700, 55, 1, 38232, 78070, 53620, 15652, 1960, 91, 1, 593859, 1301088, 1007292, 356048, 60550, 4704, 140, 1, 10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1, 202601898, 495263284, 456715752, 209878440, 52619854, 7322172, 545908, 19800, 285, 1, 4342263000, 11085720018, 10921213644, 5516785032, 1579263840, 264576774, 25677652, 1372228, 36300, 385, 1
Offset: 0

Views

Author

Gheorghe Coserea, May 21 2017

Keywords

Examples

			A(x;t) = 1 + (1 + t)*x + (4 + 5*t + t^2)*x^2 + (27 + 40*t + 14*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]      [4]      [5]     [6]    [7]  [8]
[0]  1;
[1]  1;        1;
[2]  4,        5,        1;
[3]  27,       40,       14,       1;
[4]  248,      419,      200,      30,      1;
[5]  2830,     5308,     3124,     700,     55,      1;
[6]  38232,    78070,    53620,    15652,   1960,    91,     1;
[7]  593859,   1301088,  1007292,  356048,  60550,   4704,   140,   1;
[8]  10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1;
[9]  ...

Links

Crossrefs

Cf. A286781, A286782, A286783, A286784, A286785.

Programs

  • Mathematica
    max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1 + x*(1 + 2*t + x*t^2)*y0[x, t]^2 + t*(1 - t)*x^2*y0[x, t]^3 + 2*x^2*y0[x, t]*D[y0[x, t], x])/(1 + 2*x*t) + O[x]^n]; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t];
Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 23 2017, adapted from PARI *)
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v);
concat(apply(p->Vecrev(p), Vec(A286796_ser(11))))

Formula

A(x;t) = Sum_{n>=0} P_n(t)*x^n = v/(1-x*t*v), where v(x;t) = A286795(x;t) and P_n(t) = Sum_{k=0..n} T(n,k)*t^k.
A000699(n+1)=T(n,0), A000330(n)=T(n,n-1), A286797(n)=P_n(1) and P_n(-1)=0 for n>0.
Showing 1-5 of 5 results.

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