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 1-7 of 7 results.

A302702 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^n)^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 2, 12, 120, 1595, 25823, 485254, 10278756, 240814116, 6159248281, 170371486813, 5060981349876, 160573684489465, 5417789356278015, 193693975380448414, 7315287863625954712, 291082028021247460862, 12174286414586563087259, 534059044249856004891501, 24524697505864740171996008
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 120*x^4 + 1595*x^5 + 25823*x^6 + 485254*x^7 + 10278756*x^8 + 240814116*x^9 + 6159248281*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 3*x^2 + 19*x^3 + 189*x^4 + 2496*x^5 + 40216*x^6 + 753775*x^7 + 15956057*x^8 + 374080591*x^9 + 6159248281*x^10 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^n)^(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, 9/3, 76/4, 945/5, 14976/6, 281512/7, 6030200/8, ...]
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  2,  12,  120,  1595,  25823,  485254, ...];
n=1: [1, 2,  5,  28,  268,  3478,  55460, 1031414, ...];
n=2: [1, 3,  9,  49,  450,  5697,  89423, 1645281, ...];
n=3: [1, 4, 14,  76,  673,  8308, 128296, 2334456, ...];
n=4: [1, 5, 20, 110,  945, 11376, 172745, 3107440, ...];
n=5: [1, 6, 27, 152, 1275, 14976, 223529, 3973746, ...];
n=6: [1, 7, 35, 203, 1673, 19194, 281512, 4944024, ...];
n=7: [1, 8, 44, 264, 2150, 24128, 347676, 6030200, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^n)^(n+1):
n=0: [1, 1,  0,   0,    0,     0,      0,       0, ...];
n=1: [1, 2,  3,   6,   29,   268,   3458,   55124, ...];
n=2: [1, 3,  9,  28,  132,  1059,  12605,  192579, ...];
n=3: [1, 4, 18,  76,  395,  2940,  31872,  459048, ...];
n=4: [1, 5, 30, 160,  945,  6986,  70100,  940180, ...];
n=5: [1, 6, 45, 290, 1950, 14976, 143807, 1796430, ...];
n=6: [1, 7, 63, 476, 3619, 29589, 281512, 3321571, ...];
n=7: [1, 8, 84, 728, 6202, 54600, 529116, 6030200, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A303062, A302703, A360234, A360235, A360236, A360237.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1,n, A=concat(A,0); A[m+1] = (Vec((1+x*Ser(A)^m)^(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)^(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)^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 1.256334309718765863868089027485828533429844901971596190707510781..., c = 0.080161548550419985236395573058502044572123359124998971614... - Vaclav Kotesovec, Oct 06 2020, updated Feb 05 2023

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

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

A360237 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+5))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 7, 87, 1495, 31865, 793769, 22290228, 689397657, 23116772771, 831159921411, 31787496335409, 1285410740283302, 54708408148614317, 2441969507507612684, 113988651908380638224, 5551479742274622439616, 281540748098045175486249, 14843765603832700589293465
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Comments

Sequences with g.f. A(x,k) such that [x^n] A(x,k)^(n+1) = [x^n] (1 + x*A(x,k)^(n+k))^(n+1) have a rate of growth: a(n) ~ c(k) * d^n * n! * n^alfa(k), where d = A360279 = 2.1246065836242897918278825746989... (independent on k) and alfa(k) = 1.256334309718765863868089027485828533429844901971596190707510781... + k*0.94499537543174739679595101598799077876961098786349034... - Vaclav Kotesovec, Feb 05 2023

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 87*x^3 + 1495*x^4 + 31865*x^5 + 793769*x^6 + 22290228*x^7 + 689397657*x^8 + 23116772771*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 + 109*x^3 + 1984*x^4 + 43816*x^5 + 1116182*x^6 + 31810516*x^7 + 994086874*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+5))^(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, 436/4, 9920/5, 262896/6, 7813274/7, 254484128/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  7,   87,  1495,  31865,  793769,  22290228, ...];
n=1: [1, 2, 15,  188,  3213,  67938, 1679767,  46874234, ...];
n=2: [1, 3, 24,  304,  5175, 108627, 2666476,  73945899, ...];
n=3: [1, 4, 34,  436,  7403, 154368, 3763020, 103713764, ...];
n=4: [1, 5, 45,  585,  9920, 205626, 4979200, 136401955, ...];
n=5: [1, 6, 57,  752, 12750, 262896, 6325530, 172251150, ...];
n=6: [1, 7, 70,  938, 15918, 326704, 7813274, 211519589, ...];
n=7: [1, 8, 84, 1144, 19450, 397608, 9454484, 254484128, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+5))^(n+1):
n=0: [1, 1,   5,   45,   585,   9920,   205626,   4979200, ...];
n=1: [1, 2,  13,  126,  1654,  27688,   563565,  13415580, ...];
n=2: [1, 3,  24,  253,  3402,  56679,  1135813,  26574702, ...];
n=3: [1, 4,  38,  436,  6065, 101400,  2008616,  46226504, ...];
n=4: [1, 5,  55,  685,  9920, 167686,  3299580,  74828790, ...];
n=5: [1, 6,  75, 1010, 15285, 262896,  5165838, 115758780, ...];
n=6: [1, 7,  98, 1421, 22519, 396109,  7813274, 173599042, ...];
n=7: [1, 8, 124, 1928, 32022, 578320, 11506804, 254484128, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360235, A360236.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+5))^(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+5))^(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+5))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.981311186877502847847844107425..., c = 0.000055660090340764345672306890127... - 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

A360236 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 6, 66, 1028, 20138, 464863, 12162876, 351915528, 11075859686, 374858234365, 13530279602015, 517628371405448, 20890826296067329, 886175281852068632, 39393952245422498344, 1830781283537184304756, 88768944166701791039297, 4482797026386165709436753, 235417696462456105986818505
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 66*x^3 + 1028*x^4 + 20138*x^5 + 464863*x^6 + 12162876*x^7 + 351915528*x^8 + 11075859686*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 + 85*x^3 + 1401*x^4 + 28339*x^5 + 666638*x^6 + 17651052*x^7 + 514911165*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+4))^(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, 340/4, 7005/5, 170034/6, 4666466/7, 141208416/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  6,  66,  1028,  20138,  464863,  12162876, ...];
n=1: [1, 2, 13, 144,  2224,  43124,  986694,  25632830, ...];
n=2: [1, 3, 21, 235,  3606,  69264, 1571169,  40527480, ...];
n=3: [1, 4, 30, 340,  5193,  98888, 2224444,  56974172, ...];
n=4: [1, 5, 40, 460,  7005, 132351, 2953185,  75110670, ...];
n=5: [1, 6, 51, 596,  9063, 170034, 3764599,  95085882, ...];
n=6: [1, 7, 63, 749, 11389, 212345, 4666466, 117060623, ...];
n=7: [1, 8, 76, 920, 14006, 259720, 5667172, 141208416, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+4))^(n+1):
n=0: [1, 1,   4,   30,   340,   5193,   98888,   2224444, ...];
n=1: [1, 2,  11,   90,  1025,  15330,  284912,   6277922, ...];
n=2: [1, 3,  21,  190,  2220,  32862,  597579,  12884601, ...];
n=3: [1, 4,  34,  340,  4131,  61208, 1094268,  23093756, ...];
n=4: [1, 5,  50,  550,  7005, 104951, 1856360,  38416740, ...];
n=5: [1, 6,  69,  830, 11130, 170034, 2996425,  61005672, ...];
n=6: [1, 7,  91, 1190, 16835, 263956, 4666466,  93880165, ...];
n=7: [1, 8, 116, 1640, 24490, 395968, 7067220, 141208416, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360235, A360237.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+4))^(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+4))^(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+4))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.036315811445755451051893091437..., c = 0.000317937301879544729612100255927... - Vaclav Kotesovec, Jan 31 2023
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