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A136632 a(n) = Sum_{k=0..n} A136630(n,k) * 2^(nk). 7

%I #8 Jan 17 2018 02:49:28

%S 1,2,16,520,66560,33882144,69055086592,564152735105152,

%T 18462508115518554112,2418626436468567646929408,

%U 1267795674038260517176495570944,2658560573512321601282555747644737536

%N a(n) = Sum_{k=0..n} A136630(n,k) * 2^(nk).

%H Robert Israel, <a href="/A136632/b136632.txt">Table of n, a(n) for n = 0..57</a>

%F From _Paul D. Hanna_, Nov 25 2009: (Start)

%F E.g.f.: Sum_{n>=0} sinh(2^n*x)^n/n!.

%F a(n) = [x^n/n! ] exp(2^n*sinh(x)).

%F (End)

%e From _Paul D. Hanna_, Nov 25 2009: (Start)

%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! +...

%e A(x) = 1 + sinh(2*x) + sinh(4*x)^2/2! + sinh(8*x)^3/3! + sinh(16*x)^4/4! +...+ sinh(2^n*x)^n/n! +...

%e a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(sinh(x)):

%e G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 12*x^5/5! + 37*x^6/6! +...+ A003724(n)*x^n/n! +... (End)

%p N:= 20: # to get a(0)..a(N)

%p E:= add(sinh(2^n*x)^n/n!,n=0..N):

%p S:= series(E,x,N+1):

%p seq(coeff(S,x,j)*j!,j=0..N); # _Robert Israel_, Jan 17 2018

%o (PARI) {a(n)=sum(k=0,n,2^(n*k)*polcoeff(x^k/prod(j=0,k\2,1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)),n))}

%o (PARI) {a(n)=n!*polcoeff(sum(k=0,n,sinh(2^k*x +x*O(x^n))^k/k!),n)} \\ _Paul D. Hanna_, Nov 25 2009

%o (PARI) {a(n)=n!*polcoeff(exp(2^n*sinh(x +x*O(x^n))),n)} \\ _Paul D. Hanna_, Nov 25 2009

%Y Cf. A136630, A003724 (row sums of A136630).

%Y Cf. A003724 (exp(sinh x)). [From _Paul D. Hanna_, Nov 25 2009]

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 14 2008

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)