OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
EXAMPLE
G.f.: 1 = ... (((((((((1/(1-x) - 1*0!*x)^2 - 2*1!*x^2)^3 - 3*2!*x^3)^4 - 6*3!*x^4)^5 - 10*4!*x^5)^6 - 27*5!*x^6)^7 - 63*6!*x^7)^8 - 290*7!*x^8)^9 - 732*8!*x^9)^10 -...- a(n)*(n-1)!*x^n )^(n+1) -...
ILLUSTRATION OF GENERATING METHOD.
Start with G1 = 1/(1-x), and proceed as follows:
G2 = (G1 - 1*0!*x)^2 = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 +...
G3 = (G2 - 2*1!*x^2)^3 = 1 + 6*x^3 + 9*x^4 + 12*x^5 + 27*x^6 + 54*x^7 + 96*x^8 +...
G4 = (G3 - 3*2!*x^3)^4 = 1 + 36*x^4 + 48*x^5 + 108*x^6 + 216*x^7 + 870*x^8 +...
G5 = (G4 - 6*3!*x^4)^5 = 1 + 240*x^5 + 540*x^6 + 1080*x^7 + 4350*x^8 + 9760*x^9 +...
G6 = (G5 - 10*4!*x^5)^6 = 1 + 3240*x^6 + 6480*x^7 + 26100*x^8 + 58560*x^9 +...
G7 = (G6 - 27*5!*x^6)^7 = 1 + 45360*x^7 + 182700*x^8 + 409920*x^9 + 1990800*x^10 +...
G8 = (G7 - 63*6!*x^7)^8 = 1 + 1461600*x^8 + 3279360*x^9 + 15926400*x^10 +...
G9 = (G8 - 290*7!*x^8)^9 = 1 + 29514240*x^9 + 143337600*x^10 + 487347840*x^11 +...
G10 = (G9 - 732*8!*x^9)^10 = 1 + 1433376000*x^10 + 4873478400*x^11 + 36114724800*x^12 +...
...
G_{n+1} = (G_{n} - a(n)*(n-1)!*x^n)^(n+1) = 1 + a(n+1)*n!*x^(n+1) +...
...
PROG
(PARI) {a(n) = my(A=[1], G = 1/(1-x +x^2*O(x^n)));
for(m=1, n-1, A=concat(A, 0); G = (G - A[m]*(m-1)!*x^m)^(m+1);
A[m+1] = polcoeff(G, m+1)/m!; ); A[n]}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Informal quick print of the initial N terms: */
{N=100; A=[1]; G = 1/(1-x +x^2*O(x^N));
for(m=1, N-1, A=concat(A, 0); G = (G - A[m]*(m-1)!*x^m)^(m+1);
A[m+1] = polcoeff(G, m+1)/m!; print1(A[m], ", "); ); print1(A[N], ", ")}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 12 2016
STATUS
approved