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A003130
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Impedances of an n-terminal network.
(Formerly M4873)
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3
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1, 12, 157, 1750, 17446, 164108, 1505099, 13720902, 125782441, 1167813944, 11029947952, 106273227216, 1046320856673, 10537366304920, 108606982421301, 1145873284492738, 12375688888657414, 136802023177966948, 1547385154016264531
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OFFSET
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2,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A003128, A003129.
Sequence in context: A110216 A218839 A036276 * A015000 A220225 A213376
Adjacent sequences: A003127 A003128 A003129 * A003131 A003132 A003133
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Sean A. Irvine, Feb 03 2015
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STATUS
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approved
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