A011556
Number of regular triangulations of triangle X n-simplex.
Original entry on oeis.org
- Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994, pp. 243-250.
A345102
a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).
Original entry on oeis.org
1, 4, 37, 589, 13276, 386059, 13741057, 578451514, 28109736811, 1548565036789, 95365652263102, 6492034471389889, 484086370908869821, 39238367740327468444, 3435176518078688461297, 323029539924876486293089, 32472511993953383052630556, 3475005417300807667690138399
Offset: 0
-
a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[7 - 6 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 3^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]
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N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(7-6*exp(x)))) \\ Seiichi Manyama, Oct 20 2021
A292783
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/sqrt(1 - 2*k*x).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 12, 15, 0, 1, 4, 27, 120, 105, 0, 1, 5, 48, 405, 1680, 945, 0, 1, 6, 75, 960, 8505, 30240, 10395, 0, 1, 7, 108, 1875, 26880, 229635, 665280, 135135, 0, 1, 8, 147, 3240, 65625, 967680, 7577955, 17297280, 2027025, 0, 1, 9, 192, 5145, 136080, 2953125, 42577920, 295540245, 518918400, 34459425, 0
Offset: 0
E.g.f. of column k: A_k(x) = 1 + k*x/1! + 3*k^2*x^2/2! + 15*k^3*x^3/3! + 105*k^4*x^4/4! + 945*k^5*x^5/5! + 10395*k^6*x^6/6! +
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 12, 27, 48, 75, ...
0, 15, 120, 405, 960, 1875, ...
0, 105, 1680, 8505, 26880, 65625, ...
0, 945, 30240, 229635, 967680, 2953125, ...
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Table[Function[k, n! SeriesCoefficient[1/Sqrt[1 - 2 k x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
A069736
Total number of Eulerian circuits in labeled multigraphs with n edges.
Original entry on oeis.org
1, 2, 13, 150, 2541, 57330, 1623105, 55405350, 2216439225, 101738006370, 5271938032725, 304455567165750, 19391501988260325, 1350480167457671250, 102096314890336391625, 8327231070135771543750, 728877648485930118800625
Offset: 0
- B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160.
- B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160.
- Valery Liskovets, A Note on the Total Number of Double Eulerian Circuits in Multigraphs , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.5
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Table[(2n)!/(2^n n!)(3^(n+1)-1)/(2(n+1)),{n,0,20}] (* Harvey P. Dale, Aug 24 2019 *)
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x=xx+O(xx^33); Vec(serlaplace((sqrt(1-2*x)-sqrt(1-6*x))/(2*x))) \\ Michel Marcus, Dec 11 2014