OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3/3! + 54*x^4/4! + 546*x^5/5! + 7644*x^6/6! + 140388*x^7/7! + 3253608*x^8/8! + 92429592*x^9/9! + 3147053520*x^10/10! + 126146938608*x^11/11! + 5866848879168*x^12/12! + 312780729436704*x^13/13! + 18921429038592288*x^14/14! + 1287533798347045536*x^15/15! +...
such that
A(x) = 1 + Integral B(x)^2 dx,
B(x) = 1 + Integral C(x)^3 dx,
C(x) = 1 + Integral D(x)^4 dx,
D(x) = 1 + Integral E(x)^5 dx,
E(x) = 1 + Integral F(x)^6 dx,
F(x) = 1 + Integral G(x)^7 dx, ...
The coefficients in these series begin:
A: [1, 1, 2, 8, 54, 546, 7644, 140388, 3253608, 92429592, ...];
B: [1, 1, 3, 18, 174, 2412, 44652, 1052664, 30551760, 1064478696, ...];
C: [1, 1, 4, 32, 404, 7164, 166560, 4852440, 171572760, 7190293320, ...];
D: [1, 1, 5, 50, 780, 16890, 474390, 16535460, 693410580, 34189099680, ...];
E: [1, 1, 6, 72, 1338, 34254, 1129596, 45937884, 2234626128, 127127805168, ...];
F: [1, 1, 7, 98, 2114, 62496, 2368464, 110207328, 6109240368, 394581185712, ...];
G: [1, 1, 8, 128, 3144, 105432, 4516512, 236792304, 14746211280, 1067014500336, ...];
H: [1, 1, 9, 162, 4464, 167454, 8002890, 466950060, 32289796260, 2588975822520, ...];
I: [1, 1, 10, 200, 6110, 253530, 13374780, 859772820, 65386201560, 5756311080360, ...]; ...
DERIVATIVES.
To illustrate a(n) = d^n/dx^n A(x) at x=0, take successive derivatives of A=A(x):
A' = B^2;
A'' = 2*B*C^3;
A''' = 2*C^6 + 6*B*C^2*D^4;
A'''' = 18*C^5*D^4 + 12*B*C*D^8 + 24*B*C^2*D^3*E^5;
A''''' = 90*C^4*D^8 + 72*C^5*D^3*E^5 + 12*C^4*D^8 + 12*B*D^12 + 96*B*C*D^7*E^5 + 24*C^5*D^3*E^5 + 48*B*C*D^7*E^5 + 72*B*C^2*D^2*E^10 + 120*B*C^2*D^3*E^4*F^6;
A'''''' = 360*C^3*D^12 + 720*C^4*D^7*E^5 + 360*C^4*D^7*E^5 + 216*C^5*D^2*E^10 + 360*C^5*D^3*E^4*F^6 + 48*C^3*D^12 + 96*C^4*D^7*E^5 + 12*C^3*D^12 + 144*B*D^11*E^5 + 96*C^4*D^7*E^5 + 96*B*D^11*E^5 + 672*B*C*D^6*E^10 + 480*B*C*D^7*E^4*F^6 + 120*C^4*D^7*E^5 + 72*C^5*D^2*E^10 + 120*C^5*D^3*E^4*F^6 + 48*C^4*D^7*E^5 + 48*B*D^11*E^5 + 336*B*C*D^6*E^10 + 240*B*C*D^7*E^4*F^6 + 72*C^5*D^2*E^10 + 144*B*C*D^6*E^10 + 144*B*C^2*D*E^15 + 720*B*C^2*D^2*E^9*F^6 + 120*C^5*D^3*E^4*F^6 + 240*B*C*D^7*E^4*F^6 + 360*B*C^2*D^2*E^9*F^6 + 480*B*C^2*D^3*E^3*F^12 + 720*B*C^2*D^3*E^4*F^5*G^7; ...
and then evaluate at x=0, where 1=A(0)=B(0)=C(0)=D(0)=E(0)=...
PROG
(PARI) {a(n) = my(A=1); for(k=0, n-1, A = 1 + intformal((A+x*O(x^n))^(n+1-k))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Print table of related series A, B, C, D, E, ... */
{a(n, r=1) = my(A=vector(3*n+2*r+2, i, 1+x));
for(m=1, 2*n+r, for(j=0, n+r+m, A[n+r+m-j+1] = 1 + intformal((A[n+r+m-j+2] + x^r*O(x^n))^(n+r+m-j+2)) ); ); polcoeff(A[r], n)}
for(r=1, 10, for(n=0, 10, print1(n!*a(n, r), ", ")); print(""))
/* Print this sequence (at row r=1): */
for(n=0, 25, print1(n!*a(n, 1), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 22 2013
STATUS
approved