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.

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A360343 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 3, 31, 526, 11907, 328980, 10580531, 384937042, 15549217485, 688430225102, 33096289502982, 1715499922758709, 95339852384471586, 5655337634718941111, 356683962066445400017, 23840465113068534382248, 1683771696557415075462436, 125327912444852044066759399
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 526*x^4 + 11907*x^5 + 328980*x^6 + 10580531*x^7 + 384937042*x^8 + 15549217485*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 4*x^2 + 41*x^3 + 687*x^4 + 15433*x^5 + 424524*x^6 + 13620842*x^7 + 495005025*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that {b(n)} begins:
[1/1, 2/2, 12/3, 164/4, 3435/5, 92598/6, 2971668/7, 108966736/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
  n=0: [1, 1,  3,  31,  526,  11907,  328980,  10580531, ...];
  n=1: [1, 2,  7,  68, 1123,  25052,  685891,  21923076, ...];
  n=2: [1, 3, 12, 112, 1800,  39555, 1072896,  34076544, ...];
  n=3: [1, 4, 18, 164, 2567,  55548, 1492336,  47093172, ...];
  n=4: [1, 5, 25, 225, 3435,  73176, 1946745,  61028770, ...];
  n=5: [1, 6, 33, 296, 4416,  92598, 2438866,  75942984, ...];
  n=6: [1, 7, 42, 378, 5523, 113988, 2971668,  91899578, ...];
  n=7: [1, 8, 52, 472, 6770, 137536, 3548364, 108966736, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n-1))^(n+1):
  n=0: [1, 1,  -1,   -2,   -26,   -463,  -10778,   -303048, ...];
  n=1: [1, 2,   3,    8,    69,   1120,   24937,    683012, ...];
  n=2: [1, 3,  12,   55,   444,   6351,  132492,   3504654, ...];
  n=3: [1, 4,  26,  164,  1411,  18560,  357624,   9024812, ...];
  n=4: [1, 5,  45,  360,  3435,  43926,  785715,  18700710, ...];
  n=5: [1, 6,  69,  668,  7134,  92598, 1570420,  35086104, ...];
  n=6: [1, 7,  98, 1113, 13279, 179816, 2971668,  62645353, ...];
  n=7: [1, 8, 132, 1720, 22794, 327032, 5403036, 108966736, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360342, A360344, A360345, A360346, A360347.
Cf. A360231, A302702.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m-1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 0.5984002265754..., c = 0.08321697608093... - Vaclav Kotesovec, Feb 06 2023

A303062 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

Original entry on oeis.org

1, 1, 2, 8, 60, 643, 8564, 133890, 2376261, 46832442, 1009739331, 23564025488, 590503218735, 15793704933899, 448695132962248, 13487808514722460, 427624923581550100, 14260707306806609885, 499071020445057149835, 18290961984686434723480, 700757535935308305865473, 28017787701624063252219677
Offset: 0

Views

Author

Paul D. Hanna, Apr 17 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 643*x^5 + 8564*x^6 + 133890*x^7 + 2376261*x^8 + 46832442*x^9 + 1009739331*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1 + x*A(x)^(n-1))^n / A(x)^n begins:
n=1: [1, 0, -2, -6, -50, -565, -7731, -122983, ...];
n=2: [1, 0, -2, -10, -89, -1030, -14307, -230054, ...];
n=3: [1, 0, 0, -9, -111, -1380, -19677, -320958, ...];
n=4: [1, 0, 4, 0, -94, -1520, -23388, -392776, ...];
n=5: [1, 0, 10, 20, 0, -1210, -24030, -436250, ...];
n=6: [1, 0, 18, 54, 225, 0, -18345, -427944, ...];
n=7: [1, 0, 28, 105, 651, 2835, 0, -316344, ...];
n=8: [1, 0, 40, 176, 1364, 8360, 41976, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

Links

Crossrefs

Cf. A303063, A302702.

Programs

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

Formula

a(n) ~ c * n! * n^(2*LambertW(1)) / LambertW(1)^n, where c = 0.153879081661359639962985708... - Vaclav Kotesovec, Aug 11 2021

A360336 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 6, 99, 2608, 90800, 3835458, 187727106, 10356030404, 632391914502, 42217751766193, 3053486035335835, 237640678130730437, 19794116975373467259, 1756875217029906875379, 165552614838271944281933, 16509692094523556884973416, 1737510282985845400007263814
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 99*x^3 + 2608*x^4 + 90800*x^5 + 3835458*x^6 + 187727106*x^7 + 10356030404*x^8 + 632391914502*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 7*x^2 + 118*x^3 + 3113*x^4 + 108221*x^5 + 4564720*x^6 + 223208259*x^7 + 12307249017*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(3*n))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 21/3, 472/4, 15565/5, 649326/6, 31953040/7, 1785666072/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  6,   99,  2608,  90800,  3835458,  187727106, ...];
n=1: [1, 2, 13,  210,  5450, 188004,  7893613,  384731112, ...];
n=2: [1, 3, 21,  334,  8544, 292017, 12186069,  591418401, ...];
n=3: [1, 4, 30,  472, 11909, 403268, 16725042,  808213780, ...];
n=4: [1, 5, 40,  625, 15565, 522211, 21523390, 1035561335, ...];
n=5: [1, 6, 51,  794, 19533, 649326, 26594644, 1273925322, ...];
n=6: [1, 7, 63,  980, 23835, 785120, 31953040, 1523791095, ...];
n=7: [1, 8, 76, 1184, 28494, 930128, 37613552, 1785666072, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(3*n))^(n+1):
n=0: [1, 1,   0,    0,     0,       0,        0,          0, ...];
n=1: [1, 2,   7,   48,   719,   17882,   603567,   25021464, ...];
n=2: [1, 3,  21,  190,  2814,   65460,  2105997,   84726534, ...];
n=3: [1, 4,  42,  472,  7303,  162828,  4982706,  193437168, ...];
n=4: [1, 5,  70,  940, 15565,  341796, 10002300,  373126910, ...];
n=5: [1, 6, 105, 1640, 29340,  649326, 18377374,  658075230, ...];
n=6: [1, 7, 147, 2618, 50729, 1150968, 31953040, 1101647800, ...];
n=7: [1, 8, 196, 3920, 82194, 1934296, 53433184, 1785666072, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360337, A360338, A302702, A360344.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(3*m))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(3*n))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.0541176773983..., c = 0.03951220887392... - Vaclav Kotesovec, Feb 06 2023
Previous Showing 11-13 of 13 results.

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