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%I M4210 #67 Nov 04 2022 07:31:36
%S 0,0,1,6,31,160,856,4802,28337,175896,1146931,7841108,56089804,
%T 418952508,3261082917,26403700954,221981169447,1934688328192,
%U 17454004213180,162765041827846,1566915224106221,15553364227949564,159004783733999787,1672432865100333916
%N Number of driving-point 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 N. J. A. Sloane, <a href="/A003128/b003128.txt">Table of n, a(n) for n = 0..100</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.
%H R. Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
%F a(n) = (Bell(n) - 3*Bell(n+1) + Bell(n+2))/2. - _Vladeta Jovovic_, Aug 07 2006
%F a(n+2) = A123158(n,4). - _Philippe Deléham_, Oct 06 2006
%F From _Peter Bala_, Nov 28 2011: (Start)
%F a(n) = Sum_{k=1..n} binomial(k,2)*Stirling2(n,k), Stirling transform of A000217.
%F a(n) = (1/(2*exp(1)))*Sum_{k>=0} k^n*(k^2-3*k+1)/k!. Note that k^2-3*k+1 = k*(k-1)-2*k+1 is an example of a Poisson-Charlier polynomial.
%F a(n) = D^n(x^2/2!*exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A005493.
%F E.g.f.: (1/2)*exp(exp(x)-1)*(exp(x)-1)^2 = x^2/2! + 6*x^3/3! + 31*x^4/4! + ...
%F O.g.f.: Sum_{k>=0} binomial(k,2)*x^k/Product_{i=1..k} (1-i*x) = x^2 + 6*x^3 + 31*x^4 + ... (End)
%F a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - 3*LambertW(n)/n). - _Vaclav Kotesovec_, Jul 28 2021
%p with(combinat); A000110:=n->sum(stirling2(n, k), k=0..n): f:=n->(A000110(n)-3*A000110(n+1)+A000110(n+2))/2;
%t a[n_] := (BellB[n] - 3*BellB[n+1] + BellB[n+2])/2; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jul 12 2012, after _Vladeta Jovovic_ *)
%t max = 23; CoefficientList[ Series[1/2*(E^x - 1)^2*E^(E^x - 1), {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Oct 04 2013, after e.g.f. *)
%o (Maxima) makelist((belln(n)-3*belln(n+1)+belln(n+2))/2,n,0,12); /* Emanuele Munarini, Jul 14 2011 */
%o (Haskell)
%o a003128 n = a003128_list !! n
%o a003128_list = zipWith3 (\x y z -> (x - 3 * y + z) `div` 2)
%o a000110_list (tail a000110_list) (drop 2 a000110_list)
%o -- _Reinhard Zumkeller_, Jun 30 2013
%o (Python)
%o # Python 3.2 or higher required
%o from itertools import accumulate
%o A003128_list, blist, a, b = [], [1], 1, 1
%o for _ in range(30):
%o ....blist = list(accumulate([b]+blist))
%o ....c = blist[-1]
%o ....A003128_list.append((c+a-3*b)//2)
%o ....a, b = b, c # _Chai Wah Wu_, Sep 19 2014
%o (Magma) [(Bell(n) - 3*Bell(n+1) + Bell(n+2))/2: n in [0..30]]; // _Vincenzo Librandi_, Sep 19 2014
%o (PARI) a(n)=sum(k=1,n,binomial(k,2)*stirling(n,k,2)) \\ _Charles R Greathouse IV_, Feb 07 2017
%o (SageMath)
%o def A003128(n): return (bell_number(n) - 3*bell_number(n+1) + bell_number(n+2))/2
%o [A003128(n) for n in range(40)] # _G. C. Greubel_, Nov 04 2022
%Y Cf. A000110, A000217, A003129, A003130, A005493, A039759, A039765, A123158.
%K nonn,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Apr 14 2000
%E Typo in entries corrected by _Martin Larsen_, Jul 03 2008
%E Typo in e.g.f. corrected by _Vaclav Kotesovec_, Feb 15 2015
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