A360581 -id:A360581 - cp's OEIS frontend

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

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

A360582 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.

Original entry on oeis.org

1, 1, 2, 8, 48, 382, 3793, 45208, 627957, 9928646, 175476102, 3420270423, 72789704826, 1678446235555, 41675807453127, 1108522434288617, 31444611938560078, 947522959703143140, 30225484159719768548, 1017558928058932606182, 36053690169955373601165, 1341103168079733579768368

Offset: 0

Paul D. Hanna, Mar 12 2023

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 382*x^5 + 3793*x^6 + 45208*x^7 + 627957*x^8 + 9928646*x^9 + 175476102*x^10 + ...
The table of coefficients in the successive powers of g.f. A(x) begins:
n = 1: [1, 1,  2,   8,   48,  382,  3793,  45208, ...];
n = 2: [1, 2,  5,  20,  116,  892,  8606, 100298, ...];
n = 3: [1, 3,  9,  37,  210, 1566, 14687, 167280, ...];
n = 4: [1, 4, 14,  60,  337, 2448, 22340, 248580, ...];
n = 5: [1, 5, 20,  90,  505, 3591, 31935, 347120, ...];
n = 6: [1, 6, 27, 128,  723, 5058, 43919, 466410, ...];
n = 7: [1, 7, 35, 175, 1001, 6923, 58828, 610653, ...];
...
The table of coefficients in A(x)/(1 + x*A(x)^n) begins:
n = 1: [1, 0,  1,   5,  34,  293, 3066, 37900, ...];
n = 2: [1, 0,  0,   3,  25,  235, 2601, 33346, ...];
n = 3: [1, 0, -1,   0,  14,  167, 2055, 28049, ...];
n = 4: [1, 0, -2,  -4,   0,   89, 1432, 21994, ...];
n = 5: [1, 0, -3,  -9, -18,    0,  742, 15216, ...];
n = 6: [1, 0, -4, -15, -41, -102,    0,  7820, ...];
n = 7: [1, 0, -5, -22, -70, -220, -775,     0, ...];
...
in which the diagonal of all zeros illustrates that
[x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.

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

Cf. A360581, A360583, A360584.

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

a(n) ~ c * n! * n^(2*LambertW(1) - 1) / LambertW(1)^n, where c = 0.11249164340900724981958... - Vaclav Kotesovec, Mar 13 2023

cp's OEIS Frontend

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

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