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.

A302703 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+1))^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 3, 21, 235, 3470, 61933, 1274893, 29423331, 747440115, 20636072811, 613611700946, 19517927805840, 660667692682175, 23699856058131981, 897955765812058192, 35832679277251514074, 1502303284645831488072, 66031982339561373164915, 3036884343153028302140119, 145885192794643951791449387
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 235*x^4 + 3470*x^5 + 61933*x^6 + 1274893*x^7 + 29423331*x^8 + 747440115*x^9 + 20636072811*x^10 + ...
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 + 31*x^3 + 356*x^4 + 5291*x^5 + 94592*x^6 + 1948763*x^7 + 45025516*x^8 + 1145651239*x^9 + 31696223593*x^10 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1, 2/2, 12/3, 124/4, 1780/5, 31746/6, 662144/7, 15590104/8, ...]
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  3,  21,  235,  3470,  61933,  1274893, ...];
n=1: [1, 2,  7,  48,  521,  7536, 132657,  2704342, ...];
n=2: [1, 3, 12,  82,  867, 12288, 213282,  4304877, ...];
n=3: [1, 4, 18, 124, 1283, 17828, 305056,  6094832, ...];
n=4: [1, 5, 25, 175, 1780, 24271, 409380,  8094540, ...];
n=5: [1, 6, 33, 236, 2370, 31746, 527824, 10326546, ...];
n=6: [1, 7, 42, 308, 3066, 40397, 662144, 12815839, ...];
n=7: [1, 8, 52, 392, 3882, 50384, 814300, 15590104, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+1))^(n+1):
n=0: [1, 1,  1,   3,   21,   235,    3470,    61933, ...];
n=1: [1, 2,  5,  18,  114,  1166,   16355,   283142, ...];
n=2: [1, 3, 12,  55,  354,  3372,   44463,   739917, ...];
n=3: [1, 4, 22, 124,  857,  7908,   98244,  1558788, ...];
n=4: [1, 5, 35, 235, 1780, 16501,  195980,  2955095, ...];
n=5: [1, 6, 51, 398, 3321, 31746,  368032,  5294250, ...];
n=6: [1, 7, 70, 623, 5719, 57302,  662144,  9182013, ...];
n=7: [1, 8, 92, 920, 9254, 98088, 1149804, 15590104, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A360234, A360235, A360236, A360237.
Cf. A360345, A360337.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+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) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(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)^(n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 2.2013296851505132606640400434738193121994558898350865326..., c = 0.026186121837027622395555466054900245177877028741031867... - Vaclav Kotesovec, Oct 06 2020, updated Feb 05 2023

A360231 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n-1))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 1, 6, 53, 628, 9167, 156309, 3021720, 64960004, 1532234825, 39270176511, 1085601040372, 32185085432757, 1018593646880447, 34279111177431666, 1222648239226278333, 46084480032637208699, 1830881732391546532475, 76488074741796221197580, 3352854778050665597014436
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 53*x^4 + 628*x^5 + 9167*x^6 + 156309*x^7 + 3021720*x^8 + 64960004*x^9 + 1532234825*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 2*x^2 + 10*x^3 + 86*x^4 + 1004*x^5 + 14507*x^6 + 246218*x^7 + 4753205*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n-1))^(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, 6/3, 40/4, 430/5, 6024/6, 101549/7, 1969744/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  1,   6,   53,  628,   9167,  156309, ...];
n=1: [1, 2,  3,  14,  119, 1374,  19732,  332844, ...];
n=2: [1, 3,  6,  25,  201, 2259,  31891,  531933, ...];
n=3: [1, 4, 10,  40,  303, 3308,  45870,  756192, ...];
n=4: [1, 5, 15,  60,  430, 4551,  61930, 1008565, ...];
n=5: [1, 6, 21,  86,  588, 6024,  80373, 1292370, ...];
n=6: [1, 7, 28, 119,  784, 7770, 101549, 1611352, ...];
n=7: [1, 8, 36, 160, 1026, 9840, 125864, 1969744, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n-1))^(n+1):
n=0: [1, 1, -1,   0,   -5,   -42,   -528,   -7939, ...];
n=1: [1, 2,  1,   0,    0,     0,      0,       0, ...];
n=2: [1, 3,  6,  10,   30,   207,   2266,   31824, ...];
n=3: [1, 4, 14,  40,  141,   808,   7694,  101288, ...];
n=4: [1, 5, 25, 100,  430,  2376,  19680,  235165, ...];
n=5: [1, 6, 39, 200, 1035,  6024,  45879,  490524, ...];
n=6: [1, 7, 56, 350, 2135, 13601, 101549,  988338, ...];
n=7: [1, 8, 76, 560, 3950, 27888, 213952, 1969744, ...]; ...
to see that the main diagonals of the tables are the same:
[1, 2, 6, 40, 430, 6024, 101549, 1969744, ...].

Links

Crossrefs

Cf. A302702, A302703, A360234, A360235, A360236, A360237.
Cf. A360337, A360345.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(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)^(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)^(n-1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279, alfa = 0.311338934287018467072138011497837... and c = 0.1932932528309324180094... - Vaclav Kotesovec, Feb 03 2023

A360338 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+2))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 8, 152, 4452, 169952, 7807014, 413004366, 24498135084, 1601156353073, 113923669100054, 8747479687135221, 720094655642863843, 63228142773931718867, 5897275794731167406208, 582262196337324537825772, 60678076577289308772410092, 6656827638797910274281675184
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 152*x^3 + 4452*x^4 + 169952*x^5 + 7807014*x^6 + 413004366*x^7 + 24498135084*x^8 + 1601156353073*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 9*x^2 + 177*x^3 + 5237*x^4 + 200533*x^5 + 9220635*x^6 + 487973429*x^7 + 28953420029*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(3*n+2))^(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, 27/3, 708/4, 26185/5, 1203198/6, 64544445/7, 3903787432/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  8,  152,  4452,  169952,  7807014,  413004366, ...];
n=1: [1, 2, 17,  320,  9272,  351240, 16048268,  845695400, ...];
n=2: [1, 3, 27,  505, 14484,  544512, 24744926, 1298895150, ...];
n=3: [1, 4, 38,  708, 20113,  750448, 33919144, 1773460112, ...];
n=4: [1, 5, 50,  930, 26185,  969761, 43594110, 2270282630, ...];
n=5: [1, 6, 63, 1172, 32727, 1203198, 53794085, 2790292344, ...];
n=6: [1, 7, 77, 1435, 39767, 1451541, 64544445, 3334457687, ...];
n=7: [1, 8, 92, 1720, 47334, 1715608, 75871724, 3903787432, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(3*n+2))^(n+1):
n=0: [1, 1,   2,   17,    320,    9272,    351240,   16048268, ...];
n=1: [1, 2,  11,  110,   1985,   54730,   2003692,   89482592, ...];
n=2: [1, 3,  27,  325,   5928,  157206,   5548868,  241397910, ...];
n=3: [1, 4,  50,  708,  13443,  348700,  11883916,  502177632, ...];
n=4: [1, 5,  80, 1305,  26185,  675816,  22359050,  916389110, ...];
n=5: [1, 6, 117, 2162,  46170, 1203198,  38962709, 1549794426, ...];
n=6: [1, 7, 161, 3325,  75775, 2016966,  64544445, 2498939864, ...];
n=7: [1, 8, 212, 4840, 117738, 3228152, 103075540, 3903787432, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360336, A360337, A360234, A360346.

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+2))^(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+2))^(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+2))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.7741677239019..., c = 0.0189263625965882... - Vaclav Kotesovec, Feb 06 2023

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
Showing 1-4 of 4 results.

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