This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075436 #36 Jan 11 2025 02:37:45
%S A075436 2,10,52,274,1452,7716,41064,218722,1165564,6213100,33125336,
%T A075436 176629268,941884088,5022886536,26786945232,142857244674,761881733148,
%U A075436 4063282813596,21670523246712,115574945807004,616395334890408,3287425475237496,17532874879557552
%N A075436 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 A075436 Invert transform gives the central binomial coefficients A000984.
%C A075436 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 A075436 Row sums of A075435.
%H A075436 Vincenzo Librandi, <a href="/A075436/b075436.txt">Table of n, a(n) for n = 2..200</a>
%H A075436 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 A075436 G.f.: x*(1-sqrt(1-4*x)-8*x)/(-3+16*x).
%F A075436 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 A075436 a(n) ~ 2^(4*n-4)/3^n. - _Vaclav Kotesovec_, Oct 13 2012
%F A075436 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 A075436 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 A075436 a(3) = 10 because 0 intermediate points produces 6 paths on a 3 X 3 board and 1 intermediate points produces 4 paths:
%e A075436 1 . 1
%e A075436 1 . 2 . 2
%e A075436 . . 2 . 4
%e A075436 or 6 + 4 = 10 paths in total.
%t A075436 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 A075436 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 A075436 (PARI) my(x='x+O('x^66)); Vec(x*(1-sqrt(1-4*x)-8*x)/(-3+16*x)) \\ _Joerg Arndt_, May 07 2013
%Y A075436 Cf. A075435, A000984.
%K A075436 easy,nonn
%O A075436 2,1
%A A075436 _Wouter Meeussen_, Sep 15 2002