A117692 -id:A117692 - cp's OEIS frontend

A117693 Row sums of A117692.

1, 6, 42, 27, 480, 265, 7070, 3815, 1820, 1449, 107338, 56903, 4636632, 2635061, 993850, 633919, 71014372, 42899857, 8111619802, 4675943415, 1414861448, 819657397, 113827776894, 75106291091, 41292848428

Offset: 1

Roger L. Bagula, Apr 12 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A034386, A117692.

A034386:= func< n | n eq 0 select 1 else LCM(PrimesInInterval(1, n)) >;
A117692:= func< n,k | A034386(n)^2/(A034386(k)*A034386(n-k)) >;
[(&+[A117692(n,k): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Jul 23 2023

f[n_]:= If[PrimeQ[n], n, 1];
cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A034386 *)
T[n_, k_]:= T[n,k]= cf[n]^2/(cf[k]*cf[n-k]);
a[n_]:= a[n]= Sum[T[n,k], {k,n}];
Table[a[n], {n,30}]

def A034386(n): return sloane.A002110(prime_pi(n))
def A117692(n,k): return A034386(n)^2/(A034386(k)*A034386(n-k))
def A117693(n): return sum(A117692(n,k) for k in range(1,n+1))
[A117693(n) for n in range(1,41)] # G. C. Greubel, Jul 23 2023

a(n) = Sum_{k=1..n} A117692(n, k).

Description simplified, offset corrected by the Assoc. Eds. of the OEIS, Jun 27 2010

A146205 Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n odd (n=15 in this example), is equal to half-integer values k+1/2, -[n/2]-1<=k<=[n/2].

Original entry on oeis.org

35, 35, 245, 245, 735, 735, 1225, 1225, 1225, 1225, 735, 735, 245, 245, 35, 35

Offset: 0

Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Oct 28 2008, May 04 2010

1) Closed-form expressions for sequences see Pfeifer (2010).
2) The median taken on partial sums of the simple random walk represents the market price in a simulation model wherein a single security among non-cooperating and asymetrically informed traders is traded (Pfeifer et al. 2009).
3) A146207=A146205+(0,A146206) see lemma 2 in Pfeifer (2010).

			All possible different paths (sequences of partial sums) in case of n=3:
{0,-1,-2,-3}; median=-1.5
{0,-1,-2,-1}; median=-1
{0,-1,0,-1}; median=-0.5
{0,-1,0,1}; median=0
{0,1,0,-1}; median=0
{0,1,0,1}; median=0.5
{0,1,2,1}; median=1
{0,1,2,3}; median=1.5
sequence of integers in case of n=3: 1,1,1,1

Pfeifer, C. (2010) Probability distribution of the median taken on partial sums of the simple random walk, Submitted to Stochastic Analysis and Applications

C. Pfeifer, K. Schredelseker, G. U. H. Seeber, On the negative value of information in informationally inefficient markets. Calculations for large number of traders, Eur. J. Operat. Res., 195 (1) (2009) 117-126.

A117692, A108347, A029460, A029466, A135553, A137272, A146206, A146207

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A117693 Row sums of A117692.

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A146205 Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n odd (n=15 in this example), is equal to half-integer values k+1/2, -[n/2]-1<=k<=[n/2].

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