A003130 Impedances of an n-terminal network.

1, 12, 157, 1750, 17446, 164108, 1505099, 13720902, 125782441, 1167813944, 11029947952, 106273227216, 1046320856673, 10537366304920, 108606982421301, 1145873284492738, 12375688888657414, 136802023177966948, 1547385154016264531

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 2..565
• J. Riordan, The number of impedances of an n-terminal network, Bell Syst. Tech. J., 18 (1939), 300-314.

Crossrefs

Cf. A003128, A003129.

Programs

Magma

A003128:= func< n | (&+[Binomial(k,2)*StirlingSecond(n,k): k in [0..n]]) >;
A003129:= func< n | (&+[Binomial(Binomial(k,2),2)*StirlingSecond(n,k): k in [0..n]]) >;
U:= func< n | 15*(&+[k*Binomial(k+1,5)*StirlingSecond(n,k): k in [0..n]]) >;
A003130:= func< n | A003128(n)+ 2*A003129(n) +U(n) >;
[A003130(n): n in [2..40]]; // G. C. Greubel, Nov 04 2022

Mathematica

A003128[n_]:= A003128[n]= Sum[StirlingS2[n, k]*Binomial[k, 2], {k,0,n}];
A003129[n_]:= A003129[n]= Sum[StirlingS2[n,k]*Binomial[Binomial[k,2],2], {k,0,n}];
U[n_]:= Sum[15*k*Binomial[k+1,5]*StirlingS2[n,k], {k,0,n}];
A003130[n_]:= A003128[n] +2*A003129[n] +U[n];
Table[A003130[n], {n,0,40}] (* G. C. Greubel, Nov 04 2022 *)

SageMath

def A003128(n): return sum(binomial(k,2)*stirling_number2(n,k) for k in range(n+1))
def A003129(n): return sum(binomial(binomial(k,2), 2)*stirling_number2(n,k) for k in range(n+1))
def U(n): return 15*sum(k*binomial(k+1,5)*stirling_number2(n,k) for k in range(n+1))
def A003130(n): return A003128(n) +2*A003129(n) +U(n)
[A003130(n) for n in range(2,40)] # G. C. Greubel, Nov 04 2022

Formula

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

Extensions

More terms from Sean A. Irvine, Feb 03 2015