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A003130 Impedances of an n-terminal network.

%I A003130 M4873 #18 Nov 04 2022 22:31:23
%S A003130 1,12,157,1750,17446,164108,1505099,13720902,125782441,1167813944,
%T A003130 11029947952,106273227216,1046320856673,10537366304920,
%U A003130 108606982421301,1145873284492738,12375688888657414,136802023177966948,1547385154016264531
%N A003130 Impedances of an n-terminal network.
%D A003130 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003130 G. C. Greubel, <a href="/A003130/b003130.txt">Table of n, a(n) for n = 2..565</a>
%H A003130 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 A003130 a(n) = A003128(n) + 2 * A003129(n) + U(n) where U(n) = Sum_{k=2..n} u(n) * Stirling2(n, k), and u(n) = (20(n)_4 + 10(n)_5 + (n)_6) / 8 where (n)_k = n * (n - 1) * ... * (n - k + 1) denotes the falling factorial. - _Sean A. Irvine_, Feb 03 2015
%t A003130 A003128[n_]:= A003128[n]= Sum[StirlingS2[n, k]*Binomial[k, 2], {k,0,n}];
%t A003130 A003129[n_]:= A003129[n]= Sum[StirlingS2[n,k]*Binomial[Binomial[k,2],2], {k,0,n}];
%t A003130 U[n_]:= Sum[15*k*Binomial[k+1,5]*StirlingS2[n,k], {k,0,n}];
%t A003130 A003130[n_]:= A003128[n] +2*A003129[n] +U[n];
%t A003130 Table[A003130[n], {n,0,40}] (* _G. C. Greubel_, Nov 04 2022 *)
%o A003130 (Magma)
%o A003130 A003128:= func< n | (&+[Binomial(k,2)*StirlingSecond(n,k): k in [0..n]]) >;
%o A003130 A003129:= func< n | (&+[Binomial(Binomial(k,2),2)*StirlingSecond(n,k): k in [0..n]]) >;
%o A003130 U:= func< n | 15*(&+[k*Binomial(k+1,5)*StirlingSecond(n,k): k in [0..n]]) >;
%o A003130 A003130:= func< n | A003128(n)+ 2*A003129(n) +U(n) >;
%o A003130 [A003130(n): n in [2..40]]; // _G. C. Greubel_, Nov 04 2022
%o A003130 (SageMath)
%o A003130 def A003128(n): return sum(binomial(k,2)*stirling_number2(n,k) for k in range(n+1))
%o A003130 def A003129(n): return sum(binomial(binomial(k,2), 2)*stirling_number2(n,k) for k in range(n+1))
%o A003130 def U(n): return 15*sum(k*binomial(k+1,5)*stirling_number2(n,k) for k in range(n+1))
%o A003130 def A003130(n): return A003128(n) +2*A003129(n) +U(n)
%o A003130 [A003130(n) for n in range(2,40)] # _G. C. Greubel_, Nov 04 2022
%Y A003130 Cf. A003128, A003129.
%K A003130 nonn
%O A003130 2,2
%A A003130 _N. J. A. Sloane_
%E A003130 More terms from _Sean A. Irvine_, Feb 03 2015