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.

A360234 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 4, 33, 414, 6750, 131963, 2957899, 73968136, 2027178710, 60143834893, 1914750144642, 64984397381766, 2339387034919340, 88976089246855623, 3563952072597604091, 149941204887915187568, 6610797722288579969347, 304837386103152855175255, 14675559490665539299350303
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 414*x^4 + 6750*x^5 + 131963*x^6 + 2957899*x^7 + 73968136*x^8 + 2027178710*x^9 + 60143834893*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 5*x^2 + 46*x^3 + 603*x^4 + 10011*x^5 + 197357*x^6 + 4444483*x^7 + 111520277*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(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, 15/3, 184/4, 3015/5, 60066/6, 1381499/7, 35555864/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  4,  33,  414,  6750,  131963,  2957899, ...];
n=1: [1, 2,  9,  74,  910, 14592,  281827,  6261048, ...];
n=2: [1, 3, 15, 124, 1500, 23673,  451690,  9944484, ...];
n=3: [1, 4, 22, 184, 2197, 34156,  643878, 14046740, ...];
n=4: [1, 5, 30, 255, 3015, 46221,  860965, 18610170, ...];
n=5: [1, 6, 39, 338, 3969, 60066, 1105794, 23681298, ...];
n=6: [1, 7, 49, 434, 5075, 75908, 1381499, 29311192, ...];
n=7: [1, 8, 60, 544, 6350, 93984, 1691528, 35555864, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+2))^(n+1):
n=0: [1, 1,   2,    9,    74,    910,   14592,   281827, ...];
n=1: [1, 2,   7,   36,   287,   3338,   51315,   963446, ...];
n=2: [1, 3,  15,   91,   744,   8337,  122662,  2227101, ...];
n=3: [1, 4,  26,  184,  1591,  17600,  249194,  4361112, ...];
n=4: [1, 5,  40,  325,  3015,  33656,  463710,  7824385, ...];
n=5: [1, 6,  57,  524,  5244,  60066,  816474, 13339956, ...];
n=6: [1, 7,  77,  791,  8547, 101619, 1381499, 22023891, ...];
n=7: [1, 8, 100, 1136, 13234, 164528, 2263888, 35555864, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360235, A360236, A360237.
Cf. A360346, A360338.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(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)^(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)^(n+2))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 3.146325060582260657459991059461810..., c = 0.007037477865521004701131626931596125... - Vaclav Kotesovec, Jan 31 2023

A360235 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+3))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 5, 48, 673, 12057, 256763, 6232909, 168035350, 4945380012, 157008686993, 5331606427775, 192417007138176, 7344652874314128, 295384546093569838, 12478509340848604628, 552330553975194126634, 25560514938260757190962, 1234444956694450007259989, 62114842767595821207341042
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 48*x^3 + 673*x^4 + 12057*x^5 + 256763*x^6 + 6232909*x^7 + 168035350*x^8 + 4945380012*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 6*x^2 + 64*x^3 + 946*x^4 + 17403*x^5 + 375913*x^6 + 9203150*x^7 + 249561291*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+3))^(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, 18/3, 256/4, 4730/5, 104418/6, 2631391/7, 73625200/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  5,  48,  673,  12057,  256763,  6232909, ...];
n=1: [1, 2, 11, 106, 1467,  25940,  546674, 13164522, ...];
n=2: [1, 3, 18, 175, 2397,  41868,  873317, 20861712, ...];
n=3: [1, 4, 26, 256, 3479,  60080, 1240618, 29397424, ...];
n=4: [1, 5, 35, 350, 4730,  80836, 1652870, 38851165, ...];
n=5: [1, 6, 45, 458, 6168, 104418, 2114759, 49309524, ...];
n=6: [1, 7, 56, 581, 7812, 131131, 2631391, 60866723, ...];
n=7: [1, 8, 68, 720, 9682, 161304, 3208320, 73625200, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+3))^(n+1):
n=0: [1, 1,   3,   18,   175,   2397,   41868,   873317, ...];
n=1: [1, 2,   9,   60,   580,   7678,  129842,  2642540, ...];
n=2: [1, 3,  18,  136,  1350,  17520,  287288,  5690016, ...];
n=3: [1, 4,  30,  256,  2661,  34404,  550050, 10593112, ...];
n=4: [1, 5,  45,  430,  4730,  61811,  971600, 18221525, ...];
n=5: [1, 6,  63,  668,  7815, 104418, 1629245, 29869968, ...];
n=6: [1, 7,  84,  980, 12215, 168294, 2631391, 47432554, ...];
n=7: [1, 8, 108, 1376, 18270, 261096, 4125864, 73625200, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360236, A360237.
Cf. A360347, A360338.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+3))^(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+3))^(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+3))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 4.09132043601400805425594207544980..., c = 0.00160512950354606176706886534963706... - Vaclav Kotesovec, Jan 31 2023

A360337 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 7, 124, 3446, 125706, 5540958, 282129207, 16148101259, 1020687876920, 70377734170699, 5246775452965364, 420104327765022458, 35937961751407922101, 3270668852260460283730, 315546031669853942486219, 32173855061751806476275665, 3457696770952845858846954590
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 124*x^3 + 3446*x^4 + 125706*x^5 + 5540958*x^6 + 282129207*x^7 + 16148101259*x^8 + 1020687876920*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 8*x^2 + 146*x^3 + 4083*x^4 + 149077*x^5 + 6569555*x^6 + 334401750*x^7 + 19137707066*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(3*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, 24/3, 584/4, 20415/5, 894462/6, 45986885/7, 2675214000/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  7,  124,  3446,  125706,  5540958,  282129207, ...];
n=1: [1, 2, 15,  262,  7189,  260040, 11396948,  577954822, ...];
n=2: [1, 3, 24,  415, 11250,  403521, 17583859,  888063051, ...];
n=3: [1, 4, 34,  584, 15651,  556696, 24118370, 1213065672, ...];
n=4: [1, 5, 45,  770, 20415,  720141, 31017985, 1553601145, ...];
n=5: [1, 6, 57,  974, 25566,  894462, 38301069, 1910335764, ...];
n=6: [1, 7, 70, 1197, 31129, 1080296, 45986885, 2283964852, ...];
n=7: [1, 8, 84, 1440, 37130, 1278312, 54095632, 2675214000, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(3*n+1))^(n+1):
n=0: [1, 1,   1,    7,   124,    3446,   125706,    5540958, ...];
n=1: [1, 2,   9,   76,  1252,   32742,  1150522,   49515052, ...];
n=2: [1, 3,  24,  253,  4179,  103866,  3499510,  146421240, ...];
n=3: [1, 4,  46,  584, 10061,  242520,  7836278,  317454824, ...];
n=4: [1, 5,  75, 1115, 20415,  487566, 15193230,  594390940, ...];
n=5: [1, 6, 111, 1892, 37119,  894462, 27139545, 1025356992, ...];
n=6: [1, 7, 154, 2961, 62412, 1538698, 45986885, 1682957396, ...];
n=7: [1, 8, 204, 4368, 98894, 2519232, 75032832, 2675214000, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360336, A360338, A360237, A360345, A302703.

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+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)^(3*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)^(3*n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.4141427006501..., c = 0.027880568114272... - 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.

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

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