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%I #8 Dec 09 2014 00:48:31
%S 1,1,8,176,6896,397888,30584128,2948178304,342418882688,
%T 46582810477568,7268517454045184,1279982790328858624,
%U 251155319283837571072,54344039464582833577984,12855960226911391575670784,3301167001281829056285458432,914476489427649778704952819712
%N E.g.f.: exp(8*x*G(x)^7) / G(x)^7 where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.
%F Let G(x) = 1 + x*G(x)^8 be the g.f. of A007556, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A'(x)/A(x) = G(x)^7.
%F (2) A'(x) = exp(8*x*G(x)^7).
%F (3) A(x) = exp( Integral G(x)^7 dx ).
%F (4) A(x) = exp( Sum_{n>=1} A234466(n-1)*x^n/n ), where A234466(n-1)(n) = binomial(8*n-2,n)/(7*n-1).
%F (5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251588.
%F (6) A(x) = Sum_{n>=0} A251588(n)*(x/A(x))^n/n! and
%F (7) [x^n/n!] A(x)^(n+1) = (n+1)*A251588(n),
%F where A251588(n) = 8^(n-6) * (n+1)^(n-8) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144).
%F a(n) = Sum_{k=0..n} 8^k * n!/k! * binomial(8*n-k-8, n-k) * (k-1)/(n-1) for n>1.
%F Recurrence: 7*(7*n-13)*(7*n-12)*(7*n-11)*(7*n-10)*(7*n-9)*(7*n-8)*(4096*n^6 - 66048*n^5 + 446400*n^4 - 1620808*n^3 + 3339890*n^2 - 3711613*n + 1743218)*a(n) = 128*(536870912*n^13 - 14831058944*n^12 + 188986949632*n^11 - 1471608258560*n^10 + 7817645654016*n^9 - 29941451735040*n^8 + 85134250240000*n^7 - 182149348773632*n^6 + 293626158621632*n^5 - 352753169299376*n^4 + 307548490429492*n^3 - 184675145918224*n^2 + 68635535585133*n - 11961900200250)*a(n-1) + 16777216*(4096*n^6 - 41472*n^5 + 177600*n^4 - 413768*n^3 + 556826*n^2 - 414321*n + 135135)*a(n-2). - _Vaclav Kotesovec_, Dec 07 2014
%F a(n) ~ 8^(8*(n-1)-1/2) / 7^(7*(n-1)-1/2) * n^(n-2) / exp(n-1). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + x + 8*x^2/2! + 176*x^3/3! + 6896*x^4/4! + 397888*x^5/5! +...
%e such that A(x) = exp(8*x*G(x)^7) / G(x)^7
%e where G(x) = 1 + x*G(x)^8 is the g.f. of A007556:
%e G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
%e Note that
%e A'(x) = exp(8*x*G(x)^7) = 1 + 8*x + 176*x^2/2! + 6896*x^3/3! +...
%e LOGARITHMIC DERIVATIVE.
%e The logarithm of the e.g.f. begins:
%e log(A(x)) = x + 7*x^2/2 + 77*x^3/3 + 1015*x^4/4 + 14763*x^5/5 +...
%e and so A'(x)/A(x) = G(x)^7.
%e TABLE OF POWERS OF E.G.F.
%e Form a table of coefficients of x^k/k! in A(x)^n as follows.
%e n=1: [1, 1, 8, 176, 6896, 397888, 30584128, 2948178304, ...];
%e n=2: [1, 2, 18, 400, 15584, 892896, 68217472, 6543183488, ...];
%e n=3: [1, 3, 30, 678, 26352, 1501344, 114073632, 10890011520, ...];
%e n=4: [1, 4, 44, 1016, 39512, 2241472, 169479808, 16107837568, ...];
%e n=5: [1, 5, 60, 1420, 55400, 3133560, 235931200, 22331561600, ...];
%e n=6: [1, 6, 78, 1896, 74376, 4200048, 315106128, 29713474944, ...];
%e n=7: [1, 7, 98, 2450, 96824, 5465656, 408881872, 38425052848, ...];
%e n=8: [1, 8, 120, 3088, 123152, 6957504, 519351232, 48658878080, ...]; ...
%e in which the main diagonal begins (see A251587):
%e [1, 2, 30, 1016, 55400, 4200048, 408881872, 48658878080, ...]
%e and is given by the formula:
%e [x^n/n!] A(x)^(n+1) = 8^(n-6) * (n+1)^(n-7) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144) for n>=0.
%t Flatten[{1,1,Table[Sum[8^k * n!/k! * Binomial[8*n-k-8, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* _Vaclav Kotesovec_, Dec 07 2014 *)
%o (PARI) {a(n) = local(G=1);for(i=1,n,G=1+x*G^8 +x*O(x^n)); n!*polcoeff(exp(8*x*G^7)/G^7, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 8^k * n!/k! * binomial(8*n-k-8,n-k) * (k-1)/(n-1) ))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A251588, A251668, A007556, A234466.
%Y Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251579, A251580.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 06 2014
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