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Number of transfer impedances of an n-terminal network.
(Formerly M3131)
3

%I M3131 #19 Nov 04 2022 13:37:28

%S 0,3,33,270,2025,14868,109851,827508,6397665,50932233,418175274,

%T 3542883864,30972408558,279287247333,2596195945977,24862074701208,

%U 245091667488207,2485294443056496,25903024863885465,277278282774462210

%N Number of transfer impedances of an n-terminal network.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A003129/b003129.txt">Table of n, a(n) for n = 2..570</a>

%H J. Riordan, <a href="https://archive.org/details/bstj18-2-300">The number of impedances of an n-terminal network</a>, Bell Syst. Tech. J., 18 (1939), 300-314.

%F E.g.f.: (exp(x) + 3)*(exp(x) - 1)^3*exp(exp(x) - 1)/8.

%F a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(binomial(k,2),2).

%t A003129[n_]:= A003129[n]= Sum[StirlingS2[n,k]*Binomial[Binomial[k,2],2], {k,0,n}];

%t Table[A003129[n], {n, 0, 40}] (* _G. C. Greubel_, Nov 04 2022 *)

%o (Magma) [(&+[Binomial(Binomial(k,2),2)*StirlingSecond(n,k): k in [0..n]]): n in [2..40]]; // _G. C. Greubel_, Nov 04 2022

%o (SageMath)

%o def A003129(n): return sum(binomial(binomial(k,2), 2)*stirling_number2(n,k) for k in range(n+1))

%o [A003129(n) for n in range(2,40)] # _G. C. Greubel_, Nov 04 2022

%Y Cf. A003128, A003130.

%K nonn

%O 2,2

%A _N. J. A. Sloane_