A075436 - OEIS

%I #32 May 20 2021 22:39:22

%S 2,10,52,274,1452,7716,41064,218722,1165564,6213100,33125336,

%T 176629268,941884088,5022886536,26786945232,142857244674,761881733148,

%U 4063282813596,21670523246712,115574945807004,616395334890408,3287425475237496,17532874879557552

%N Right- or upward-moving paths connecting opposite corners of an n X n chessboard, visiting the diagonal in 0 up to (n-2) intermediate points between start and finish. Equivalently, subdivide the chessboard into 1 up to (n-1) blocks along the diagonal in all possible ways and sum the path-count over all sub-blocks.

%C Invert transform gives the central binomial coefficients A000984.

%C If it is required that the paths stay at the same side of the diagonal between intermediate points, then the count of intermediate points becomes an exact count of crossings and one gets the central binomial coefficients A000984.

%C Row sums of A075435.

%H Vincenzo Librandi, <a href="/A075436/b075436.txt">Table of n, a(n) for n = 2..200</a>

%H Cyril Banderier, Markus Kuba, and Michael Wallner, <a href="https://arxiv.org/abs/2103.03751">Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions</a>, arXiv:2103.03751 [math.PR], 2021.

%F G.f.: x*(1-sqrt(1-4*x)-8*x)/(-3+16*x).

%F Recurrence (for n>3): 3*(n-1)*a(n) = 2*(14*n-23)*a(n-1)-32*(2*n-5)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012

%F a(n) ~ 2^(4*n-4)/3^n. - _Vaclav Kotesovec_, Oct 13 2012

%F a(n) = 2^(4*n-7)/3^(n-2) * (1 - Sum_{k=2..n-1} C(2*k-1,k)*3^(k-2)/((2*k-1) * 2^(4*k-4)) ), for n>2. - _Vaclav Kotesovec_, Oct 28 2012

%F G.f.: 2/(Q(0)-4*x), where Q(k) = 2*x + (k+1)/(2*k+1) - 2*x*(k+1)/(2*k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 03 2013

%e a(3) = 10 because 0 intermediate points produces 6 paths on a 3 X 3 board and 1 intermediate points produces 4 paths:

%e 1 . 1

%e 1 . 2 . 2

%e . . 2 . 4

%e or 6 + 4 = 10 paths in total.

%t Rest[CoefficientList[Series[(1-Sqrt[1-4*x]-8*x)/(-3+16*x), {x, 0, 24}], x]]  (* corrected by _Vaclav Kotesovec_, Oct 28 2012 *)  or combinatorially: Plus@@@Table[Table[Plus@@Apply[Times, Compositions[n-1-k, k]+1 /. i_Integer->Binomial[2i, i], {1}], {k, 1, n-1}], {n, 2, 12}]

%t Flatten[{2,Table[2^(4*n-7)/3^(n-2)*(1-Sum[Binomial[2*k-1,k]*3^(k-2)/((2*k-1)*2^(4*k-4)),{k,2,n-1}] ),{n,3,20}]}] (* _Vaclav Kotesovec_, Oct 28 2012 *)

%o (PARI) x='x+O('x^66); Vec(x*(1-sqrt(1-4*x)-8*x)/(-3+16*x)) \\ _Joerg Arndt_, May 07 2013

%Y Cf. A075435, A000984.

%K easy,nonn

%O 2,1

%A _Wouter Meeussen_, Sep 15 2002