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%I #9 Dec 09 2014 00:49:41
%S 1,1,7,133,4501,224497,14926387,1245099709,125177105641,
%T 14743403405857,1991987858095039,303781606238806549,
%U 51624122993243471293,9674836841745014156497,1982441139367342976694379,440946185623028320815311053,105810290178441439797537070033,27247415403508413760437930799681
%N E.g.f.: exp(7*x*G(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.
%F Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A'(x)/A(x) = G(x)^6.
%F (2) A'(x) = exp(7*x*G(x)^6).
%F (3) A(x) = exp( Integral G(x)^6 dx ).
%F (4) A(x) = exp( Sum_{n>=1} A130565(n)*x^n/n ), where A130565(n) = binomial(7*n-2,n)/(6*n-1).
%F (5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251587.
%F (6) A(x) = Sum_{n>=0} A251587(n)*(x/A(x))^n/n! and
%F (7) [x^n/n!] A(x)^(n+1) = (n+1)*A251587(n),
%F where A251587(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807).
%F a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-7, n-k) * (k-1)/(n-1) for n>1.
%F Recurrence: 72*(2*n-3)*(3*n-5)*(3*n-4)*(6*n-11)*(6*n-7)*(2401*n^5 - 32585*n^4 + 178311*n^3 - 492779*n^2 + 689623*n - 392491)*a(n) = 7*(282475249*n^11 - 6658345155*n^10 + 71339412375*n^9 - 458968749330*n^8 + 1971937124661*n^7 - 5947597074909*n^6 + 12867618998885*n^5 - 20002508046570*n^4 + 21938241804255*n^3 - 16207858252075*n^2 + 7281095411817*n - 1512276480000)*a(n-1) - 823543*(2401*n^5 - 20580*n^4 + 71981*n^3 - 129346*n^2 + 120663*n - 47520)*a(n-2). - _Vaclav Kotesovec_, Dec 07 2014
%F a(n) ~ 7^(7*(n-1)-1/2) / 6^(6*(n-1)-1/2) * n^(n-2) / exp(n-1). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
%e such that A(x) = exp(7*x*G(x)^6) / G(x)^6
%e where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:
%e G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
%e Note that
%e A'(x) = exp(7*x*G(x)^6) = 1 + 7*x + 133*x^2/2! + 4501*x^3/3! +...
%e LOGARITHMIC DERIVATIVE.
%e The logarithm of the e.g.f. begins:
%e log(A(x)) = x + 6*x^2/2 + 57*x^3/3 + 650*x^4/4 + 8184*x^5/5 + 109668*x^6/6 +...
%e and so A'(x)/A(x) = G(x)^6.
%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, 7, 133, 4501, 224497, 14926387, 1245099709, ...];
%e n=2: [1, 2, 16, 308, 10360, 512624, 33845728, 2807075264, ...];
%e n=3: [1, 3, 27, 531, 17829, 876771, 57529143, 4745597787, ...];
%e n=4: [1, 4, 40, 808, 27184, 1331008, 86864512, 7129675840, ...];
%e n=5: [1, 5, 55, 1145, 38725, 1891205, 122869075, 10038831425, ...];
%e n=6: [1, 6, 72, 1548, 52776, 2575152, 166702752, 13564381824, ...];
%e n=7: [1, 7, 91, 2023, 69685, 3402679, 219682183, 17810832319, ...];
%e n=8: [1, 8, 112, 2576, 89824, 4395776, 283295488, 22897384832, ...]; ...
%e in which the main diagonal begins (see A251587):
%e [1, 2, 27, 808, 38725, 2575152, 219682183, 22897384832, ...]
%e and is given by the formula:
%e [x^n/n!] A(x)^(n+1) = 7^(n-5) * (n+1)^(n-6) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807) for n>=0.
%t Flatten[{1,1,Table[Sum[7^k * n!/k! * Binomial[7*n-k-7, 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^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G^6, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-7,n-k) * (k-1)/(n-1) ))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A251587, A251667, A002296, A130565.
%Y Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251578, A251579, A251580.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 06 2014
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