A075435
T(n,k) = right- or upward-moving paths connecting opposite corners of an n X n chessboard, visiting the diagonal at k points between start and finish.
Original entry on oeis.org
2, 6, 4, 20, 24, 8, 70, 116, 72, 16, 252, 520, 456, 192, 32, 924, 2248, 2496, 1504, 480, 64, 3432, 9520, 12624, 9728, 4480, 1152, 128, 12870, 39796, 60792, 56400, 33440, 12480, 2688, 256, 48620, 164904, 283208, 304704, 218720, 105600, 33152, 6144
Offset: 2
{2},
{6, 4},
{20, 24, 8},
{70, 116, 72, 16},
{252, 520, 456, 192, 32},
...
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# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> binomial(2*n, n)); # Peter Luschny, Oct 19 2022
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Table[Table[Plus@@Apply[Times, Compositions[n-1-k, k]+1 /. i_Integer->Binomial[2i, i], {1}], {k, 1, n-1}], {n, 2, 12}]
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T(n,m):=sum(k/n*binomial(2*n-k-1,n-1)*2^k*binomial(k-1,m-1),k,m,n); /* Vladimir Kruchinin, Mar 30 2011 */
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@cached_function
def T(k,n):
if k==n: return 2^n
if k==0: return 0
return sum(binomial(2*i,i)*T(k-1,n-i) for i in (1..n-k+1))
A075435 = lambda n,k: T(k,n)
for n in (1..9): print([A075435(n,k) for k in (1..n)]) # Peter Luschny, Mar 12 2016
A328004
Expansion of e.g.f. 1 / (1 - Sum_{k>=1} binomial(2*k,k) * x^k / k!).
Original entry on oeis.org
1, 2, 14, 140, 1854, 30692, 609812, 14135816, 374486782, 11161030388, 369597971484, 13463177200376, 535000400076660, 23031528320070584, 1067766010124118200, 53038672987708575920, 2810204538580052967422, 158202066016882053997204, 9429962256806049820343564
Offset: 0
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nmax = 18; CoefficientList[Series[1/(2 - Exp[2 x] BesselI[0, 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(2 - exp(2*x) * (besseli(0,2*x))))) \\ Michel Marcus, Oct 02 2019
A305561
Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).
Original entry on oeis.org
1, 1, 3, 8, 23, 64, 182, 512, 1451, 4096, 11594, 32768, 92710, 262144, 741548, 2097152, 5931955, 16777216, 47454210, 134217728, 379628818, 1073741824, 3037013748, 8589934592, 24296051198, 68719476736, 194368201572, 549755813888, 1554944869676, 4398046511104
Offset: 0
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m:=35; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - Sqrt(1 - 4*x^2)))); // Vincenzo Librandi, Jan 27 2020
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a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-i)*binomial(i, floor(i/2)), i=1..n))
end:
seq(a(n), n=0..35); # Alois P. Heinz, Jun 21 2018
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nmax = 29; CoefficientList[Series[2 x (1 - 2 x)/(1 + 2 x - 8 x^2 - Sqrt[1 - 4 x^2]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[Binomial[k, Floor[k/2]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]
A370375
Number of compositions of n where there are A005809(k) sorts of part k.
Original entry on oeis.org
1, 3, 24, 201, 1710, 14649, 125934, 1084716, 9353574, 80711625, 696756420, 6016526145, 51962422464, 448833782556, 3877191573720, 33494487646632, 289365173239302, 2499947731531305, 21598513018825920, 186604716462810075, 1612224571249844910
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, binomial(3*k, k)*x^k)))
A373816
Expansion of 1/(2 - 1/(1 - 4*x)^(3/2)).
Original entry on oeis.org
1, 6, 66, 716, 7746, 83748, 905332, 9786456, 105788610, 1143539764, 12361276668, 133621178280, 1444399211188, 15613460929512, 168776165962152, 1824412555458480, 19721274932411586, 213180228200294100, 2304405260270678316, 24909831687105112968
Offset: 0
Showing 1-5 of 5 results.
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