OFFSET
1,3
FORMULA
Let A_n(x) denote the n-th iteration of g.f. A(x), then A(x) satisfies:
. A(A(x)) = Sum_{n>=0} A_{n+1}(x) * A(A(x))^n;
. A_k(x) = Sum_{n>=0} A_{n+k-1}(x) * A_k(x)^n for all k;
. A_{-1}(x) = x - Sum_{n>=1} A_{n-1}(x)*x^n = series reversion of A(x).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 56*x^5 + 290*x^6 + 1626*x^7 +...
Coefficients in the initial ITERATIONS of g.f. A(x) begin:
A_0: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
A_1: [1, 1, 3, 12, 56, 290, 1626, 9735, 61709, ...];
A_2: [1, 2, 8, 40, 226, 1386, 9040, 61977, 443399, ...];
A_3: [1, 3, 15, 90, 598, 4256, 31888, 248974, 2012670, ...];
A_4: [1, 4, 24, 168, 1284, 10416, 88352, 776634, 7033622, ...];
A_5: [1, 5, 35, 280, 2420, 22050, 209038, 2045481, 20551303, ...];
A_6: [1, 6, 48, 432, 4166, 42130, 441584, 4762787, 52608941, ...];
A_7: [1, 7, 63, 630, 6706, 74536, 855988, 10090920, 121577608, ...];
A_8: [1, 8, 80, 880, 10248, 124176, 1550656, 19836948, 258918284, ...]; ...
where the antidiagonal sums of the above table form the unsigned coefficients in A_{-1}(x), the series reversion of A(x), which begins:
A_{-1}(x) = x - x^2 - x^3 - 2*x^4 - 6*x^5 - 24*x^6 - 116*x^7 - 636*x^8 - 3820*x^9 - 24651*x^10 - 168914*x^11 - 1219501*x^12 -...
Coefficients in the initial POWERS of g.f. A(x) begin:
A^1: [1, 1, 3, 12, 56, 290, 1626, 9735, 61709, ...];
A^2: [1, 2, 7, 30, 145, 764, 4312, 25806, 162740, ...];
A^3: [1, 3, 12, 55, 276, 1485, 8469, 50847, 320031, ...];
A^4: [1, 4, 18, 88, 459, 2528, 14610, 88256, 555937, ...];
A^5: [1, 5, 25, 130, 705, 3981, 23365, 142345, 899500, ...];
A^6: [1, 6, 33, 182, 1026, 5946, 35497, 218508, 1387926, ...];
A^7: [1, 7, 42, 245, 1435, 8540, 51919, 323408, 2068290, ...];
A^8: [1, 8, 52, 320, 1946, 11896, 73712, 465184, 2999491, ...]; ...
where the leading n zeros in A(x)^n have been omitted.
Coefficients in the initial PRODUCTS A_n(x)*A(x)^n begin:
A_0*A^0: [1, 0, 0, 0, 0, 0, 0, 0, ...];
A_1*A^1: [1, 2, 7, 30, 145, 764, 4312, 25806, ...];
A_2*A^2: [1, 4, 19, 100, 567, 3412, 21594, 142881, ...];
A_3*A^3: [1, 6, 36, 226, 1489, 10268, 73846, 551969, ...];
A_4*A^4: [1, 8, 58, 424, 3199, 25052, 203650, 1715002, ...];
A_5*A^5: [1, 10, 85, 710, 6050, 53206, 484133, 4552426, ...];
A_6*A^6: [1, 12, 117, 1100, 10460, 102220, 1030887, 10720923, ...];
A_7*A^7: [1, 14, 154, 1610, 16912, 181958, 2015846, 22985229, ...];
A_8*A^8: [1, 16, 196, 2256, 25954, 304984, 3683120, 45697748, ...]; ...
where the antidiagonal sums in the above table forms this sequence.
PROG
(PARI) {a(n)=local(F=x+x^2, G); for(i=1, n, G=x; F=x+sum(k=1, n, (G=subst(G, x, F+x*O(x^n)))*F^k)); polcoeff(F, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 20 2010
STATUS
approved