A298698 G.f. A(x) satisfies: A(x) = Sum_{n>=0} Series_Reversion( x/A(x)^n )^n.

1, 1, 2, 8, 47, 350, 3067, 30323, 329573, 3870652, 48536482, 644216596, 8991863592, 131325765519, 1999067157044, 31616923907586, 518229831404178, 8784768301425245, 153742465593453290, 2773882320116359084, 51533402856121902172, 984815753721135057318, 19342564902607533291194, 390167498138209335212685, 8077917911125696744670518, 171566739311933583272242140

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Example

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 47*x^4 + 350*x^5 + 3067*x^6 + 30323*x^7 + 329573*x^8 + 3870652*x^9 + 48536482*x^10 + 644216596*x^11 + 8991863592*x^12 + ...
such that
A(x) = 1 + x*R_1(x) + x^2*R_2(x)^4 + x^3*R_3(x)^9 + x^4*R_4(x)^16 + x^5*R_5(x)^25 + x^6*R_6(x)^36 + x^7*R_7(x)^49 + ... + x^n * R_{n}(x)^(n^2) + ...
where R_{n}(x) = A( x*R_{n}(x)^n ),
and A(x) = R_{n}( x/A(x)^n ),
thus R_{n}(x)^n = Series_Reversion( x/A(x)^n ) / x.
RELATED TABLES.
The coefficients in the functions R_{n}(x) begin:
R_1(x): [1, 1, 3, 15, 100, 806, 7459, 76846, 863768, ...];
R_2(x): [1, 1, 4, 25, 197, 1802, 18371, 203960, 2430447, ...];
R_3(x): [1, 1, 5, 38, 354, 3738, 43102, 531666, 6928810, ...];
R_4(x): [1, 1, 6, 54, 587, 7139, 93703, 1301387, 18898185, ...];
R_5(x): [1, 1, 7, 73, 912, 12655, 188273, 2945597, 47899852, ...];
R_6(x): [1, 1, 8, 95, 1345, 21061, 352255, 6174257, 112111283, ...];
R_7(x): [1, 1, 9, 120, 1902, 33257, 619732, 12078058, 243394942, ...];
...
where R_{n}(x) = A( x*R_{n}(x)^n ).
The coefficients in the functions R_{n}(x)^n begin:
R_1(x)^1: [1, 1, 3, 15, 100, 806, 7459, 76846, 863768, ...];
R_2(x)^2: [1, 2, 9, 58, 460, 4198, 42547, 468928, 5544691, ...];
R_3(x)^3: [1, 3, 18, 145, 1380, 14667, 169013, 2075478, ...];
R_4(x)^4: [1, 4, 30, 292, 3285, 40592, 536356, 7459984, ...];
R_5(x)^5: [1, 5, 45, 515, 6725, 95536, 1439850, 22683150, ...];
R_6(x)^6: [1, 6, 63, 830, 12375, 199542, 3397376, 60217200, ...];
R_7(x)^7: [1, 7, 84, 1253, 21035, 380429, 7243061, 143203670, ...];
...
where R_{n}(x)^n = Series_Reversion( x/A(x)^n ) / x.
The coefficients in the functions R_{n}(x)^(n^2) begin:
R_1(x)^1: [1, 1, 3, 15, 100, 806, 7459, 76846, ...];
R_2(x)^4: [1, 4, 22, 152, 1233, 11280, 113530, 1236968, ...];
R_3(x)^9: [1, 9, 81, 786, 8208, 91332, 1073232, 13226598, ...];
R_4(x)^16: [1, 16, 216, 2864, 38572, 532112, 7528856, 109164400, ...];
R_5(x)^25: [1, 25, 475, 8325, 142250, 2419305, 41311975, ...];
R_6(x)^36: [1, 36, 918, 20640, 438705, 9078228, 185516262, ...];
R_7(x)^49: [1, 49, 1617, 45472, 1180018, 29287349, 708195040, ...];
...
where A(x) = Sum_{n>=0} x^n * R_{n}(x)^(n^2).

Mathematica

a[n_] := Module[{A = 1+x}, Do[A = Sum[InverseSeries[x/A^m + x*O[x]^n, x]^m, {m, 0, n}], {n}]; Coefficient[A, x, n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Feb 15 2018 *)

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, serreverse(x/A^m +x*O(x^n))^m )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

f. A125224.

Sequence in context: A297014 A102009 A254747 * A354498 A135904 A145846

Adjacent sequences: A298695 A298696 A298697 * A298699 A298700 A298701

Keyword

nonn

Author

Paul D. Hanna, Feb 14 2018

Status

approved