A003128 Number of driving-point impedances of an n-terminal network. (Formerly M4210)

0, 0, 1, 6, 31, 160, 856, 4802, 28337, 175896, 1146931, 7841108, 56089804, 418952508, 3261082917, 26403700954, 221981169447, 1934688328192, 17454004213180, 162765041827846, 1566915224106221, 15553364227949564, 159004783733999787, 1672432865100333916

Offset

0,4

References

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..100

J. Riordan, The number of impedances of an n-terminal network, Bell Syst. Tech. J., 18 (1939), 300-314.

R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Formula

a(n) = (Bell(n) - 3*Bell(n+1) + Bell(n+2))/2. - Vladeta Jovovic, Aug 07 2006

a(n+2) = A123158(n,4). - Philippe Deléham, Oct 06 2006

From Peter Bala, Nov 28 2011: (Start)

a(n) = Sum_{k=1..n} binomial(k,2)*Stirling2(n,k), Stirling transform of A000217.

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.

a(n) = D^n(x^2/2!*exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A005493.

E.g.f.: (1/2)*exp(exp(x)-1)*(exp(x)-1)^2 = x^2/2! + 6*x^3/3! + 31*x^4/4! + ...

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)

a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - 3*LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

Maple

with(combinat); A000110:=n->sum(stirling2(n, k), k=0..n): f:=n->(A000110(n)-3*A000110(n+1)+A000110(n+2))/2;

Mathematica

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 *)
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. *)

Prog

(Maxima) makelist((belln(n)-3*belln(n+1)+belln(n+2))/2, n, 0, 12); /* Emanuele Munarini, Jul 14 2011 */
(Haskell)
a003128 n = a003128_list !! n
a003128_list = zipWith3 (\x y z -> (x - 3 * y + z) `div` 2)
               a000110_list (tail a000110_list) (drop 2 a000110_list)
-- Reinhard Zumkeller, Jun 30 2013
(Python)
# Python 3.2 or higher required
from itertools import accumulate
A003128_list, blist, a, b = [], [1], 1, 1
for _ in range(30):
....blist = list(accumulate([b]+blist))
....c = blist[-1]
....A003128_list.append((c+a-3*b)//2)
....a, b = b, c # Chai Wah Wu, Sep 19 2014
(Magma) [(Bell(n) - 3*Bell(n+1) + Bell(n+2))/2: n in [0..30]]; // Vincenzo Librandi, Sep 19 2014
(PARI) a(n)=sum(k=1, n, binomial(k, 2)*stirling(n, k, 2)) \\ Charles R Greathouse IV, Feb 07 2017
(SageMath)
def A003128(n): return (bell_number(n) - 3*bell_number(n+1) + bell_number(n+2))/2
[A003128(n) for n in range(40)] # G. C. Greubel, Nov 04 2022

Crossrefs

Cf. A000110, A000217, A003129, A003130, A005493, A039759, A039765, A123158.

Sequence in context: A022034 A277669 A047665 * A058146 A015449 A336945

Adjacent sequences: A003125 A003126 A003127 * A003129 A003130 A003131

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 14 2000

Typo in entries corrected by Martin Larsen, Jul 03 2008

Typo in e.g.f. corrected by Vaclav Kotesovec, Feb 15 2015

Status

approved