A003130 Impedances of an n-terminal network. (Formerly M4873)

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

Offset

2,2

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.

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A003128, A003129.

Sequence in context: A110216 A218839 A036276 * A015000 A220225 A213376

Adjacent sequences: A003127 A003128 A003129 * A003131 A003132 A003133

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Feb 03 2015

Status

approved