A303062 -id:A303062 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Apr 16 2018

nonn

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..300

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

{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),", "))

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

A303063 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^n)^n / A(x)^n = 0 for n>1.

Original entry on oeis.org

1, 1, 3, 17, 151, 1812, 26766, 461302, 8978490, 193200156, 4529641423, 114510000515, 3097375627215, 89116723381943, 2714808312021989, 87242980758842543, 2948618278635037930, 104544558380661516685, 3880035778583841094470, 150451784852703095162304, 6084892588256393044757197, 256294338370540915598727500

Offset: 0

Paul D. Hanna, Apr 17 2018

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1812*x^5 + 26766*x^6 + 461302*x^7 + 8978490*x^8 + 193200156*x^9 + 4529641423*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1 + x*A(x)^n)^n / A(x)^n begins:
n=1: [1, 0, -2, -12, -116, -1475, -22625, -400078, ...];
n=2: [1, 0, -2, -18, -197, -2630, -41347, -742194, ...];
n=3: [1, 0, 0, -15, -228, -3390, -55716, -1022901, ...];
n=4: [1, 0, 4, 0, -178, -3536, -64144, -1228756, ...];
n=5: [1, 0, 10, 30, 0, -2640, -63025, -1327450, ...];
n=6: [1, 0, 18, 78, 369, 0, -45519, -1252758, ...];
n=7: [1, 0, 28, 147, 1008, 5425, 0, -881412, ...];
n=8: [1, 0, 40, 240, 2012, 15080, 91832, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + x*A(x)^n)^n / A(x)^n = 0 for n>1.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A303062, A360578, A360581.

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

a(n) ~ c * n! * n^(3*LambertW(1) + 1/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.03203091421745281863810572012... - Vaclav Kotesovec, Aug 11 2021

cp's OEIS Frontend

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

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