A290847
Number of dominating sets in the n-triangular graph.
Original entry on oeis.org
1, 7, 57, 973, 32057, 2079427, 267620753, 68649126489, 35172776136145, 36025104013571583, 73784683970720501897, 302228664636911612364581, 2475873390079769597467385417, 40564787539999607393632514635067, 1329227699017403425105119604848703905
Offset: 2
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b[n_]:=Sum[(-1)^(n - k)*Binomial[n, k]*2^Binomial[k, 2], {k, 0, n}]; a[n_]:=b[n] + n*b[n - 1]; Table[a[n], {n, 2, 20}] (* Indranil Ghosh, Aug 12 2017 *)
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\\ here b(n) is A006129
b(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
a(n) = b(n) + n*b(n-1);
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from sympy import binomial
def b(n): return sum((-1)**(n - k)*binomial(n, k)*2**binomial(k, 2) for k in range(n + 1))
def a(n): return b(n) + n*b(n - 1)
print([a(n) for n in range(2, 21)]) # Indranil Ghosh, Aug 13 2017
A303227
Number of minimal total dominating sets in the n-triangular (Johnson) graph.
Original entry on oeis.org
1, 1, 0, 3, 12, 80, 840, 4032, 31976, 371016, 4354650, 55066880, 680003412, 9047989392, 132626606294, 2096065474440, 34991505975120, 607163217989312, 11006996786618994, 209218563659672064, 4168806234781798100, 86745911047924139760, 1876774293382882814382
Offset: 0
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Range[0, 20]! CoefficientList[Series[Exp[x^3/2] + x Exp[x Exp[x^2 + x] - (x + x^2 + x^3 + x^4/2)], {x, 0, 20}], x] (* Eric W. Weisstein, Apr 23 2018 *)
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seq(n)={Vec(serlaplace(exp(x^3/2 + O(x*x^n)) + x*exp(x*exp(x^2+x + O(x^n)) - (x+x^2+x^3+x^4/2))))} \\ Andrew Howroyd, Apr 21 2018
A298104
Number of connected dominating sets in the n-triangular graph.
Original entry on oeis.org
1, 7, 54, 918, 31072, 2053184, 266478640, 68560228400, 35159451505536, 36021118923496320, 73782296097354062336, 302225812400352055040512, 2475866621867539032536216576, 40564755669895936890639118713856, 1329227401230786888692092742930946048
Offset: 2
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a[n_] := a[n] = 2^Binomial[n, 2] - Sum[k Binomial[n, k] 2^((n - k) (n - k - 1)/2) a[k], {k, n - 1}]/n; Join[{1}, Table[a[n] + n a[n - 1], {n, 3, 20}]] (* Eric W. Weisstein, Jan 15 2018 *)
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b(n)={serlaplace(-x + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k! + O(x*x^n))))}
{ my(n=20); Vec(b(n) + x*deriv(x*b(n))) } \\ Andrew Howroyd, Jan 14 2018
A323499
Number of minimum dominating sets in the n-triangular (Johnson) graph.
Original entry on oeis.org
1, 3, 15, 15, 195, 105, 2625, 945, 38745, 10395, 634095, 135135, 11486475, 2027025, 229053825, 34459425, 4996616625, 654729075, 118505962575, 13749310575, 3038597637075, 316234143225, 83802047954625, 7905853580625, 2474532170735625, 213458046676875
Offset: 2
Showing 1-4 of 4 results.
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