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 A100234 G.f. A(x) satisfies: 6^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (6+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n. 2
 1, 4, 5, -15, 20, 90, -695, 1785, 3895, -53985, 196255, 121635, -4907130, 23332140, -13181145, -470127465, 2866898820, -4455872910, -44776087145, 356263904235, -873534120380, -3988869806010, 44179467566755, -147200296896765, -293052319462105, 5409366658571715 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2. LINKS Table of n, a(n) for n=0..25. FORMULA a(n)=-(3*(2*n-3)*a(n-1)+29*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=4, a(2)=5. G.f.: A(x) = (1+5*x+sqrt(1+6*x+29*x^2))/2. EXAMPLE From the table of powers of A(x) (A100235), we see that 6^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n: A^1=[1,4],5,-15,20,90,-695,1785,... A^2=[1,8,26],10,-55,190,-245,-1690,... A^3=[1,12,63,139],15,-120,635,-2130,... A^4=[1,16,116,436,726],20,-210,1480,... A^5=[1,20,185,965,2830,3774],25,-325,... A^6=[1,24,270,1790,7335,17634,19601],30,... PROG (PARI) a(n)=if(n==0, 1, (6^n-1-sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n+x*O(x^k), k)))/n) (PARI) a(n)=if(n==0, 1, if(n==1, 4, if(n==2, 5, -(3*(2*n-3)*a(n-1)+29*(n-3)*a(n-2))/n))) (PARI) a(n)=polcoeff((1+5*x+sqrt(1+6*x+29*x^2+x^2*O(x^n)))/2, n) CROSSREFS Cf. A100235, A100236, A100231. Sequence in context: A047184 A002509 A230983 * A007390 A037955 A225121 Adjacent sequences: A100231 A100232 A100233 * A100235 A100236 A100237 KEYWORD sign AUTHOR Paul D. Hanna, Nov 29 2004 STATUS approved

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Last modified December 6 08:34 EST 2023. Contains 367600 sequences. (Running on oeis4.)