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%I #51 Sep 10 2021 19:29:47
%S 1,4,17,80,401,2084,11073,59712,325441,1788004,9885457,54932176,
%T 306528145,1716461764,9640310017,54282691840,306337928449,
%U 1732172652868,9811489710737,55660919625680,316204733423121,1798580947651044
%N Partial sums of the central Delannoy numbers (A001850).
%C Number of peaks at odd level in all Schroeder paths (i.e., consisting of steps U=(1,1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0). Example: a(1)=4 because HH,HU*D,U*DH,UHD,U*DU*D,UUDD contain 4 peaks at odd level (indicated by *).
%C From Evert-Jan D. Pol (evert-jan.pol(AT)philips.com), Oct 25 2005: "Also appears in the context of infinite lattices of unit resistors. The paper by Atkinson and van Steenwijk shows how to calculate the resulting resistance R(n,p) between two nodes in the lattice that are apart by the vector (n,p). The resulting values can be written in the form r+s/Pi, where r and s are rational numbers.
%C "Here we concentrate on the rational part r. The paper gives values for a single quadrant in the integer plane. Other quadrants can be filled with mirror-images of the given quadrant. Casual inspection of the values shows that the numbers are most easily analysed by looking at the diagonals (n+k,n-k) for n=0,1,2,... and fixed k. The rational part of the values on these diagonals appears to be a polynomial sequence of degree 2k-1, apart from the alternating sign.
%C "Similarly, the absolute value of the rational part of the values on the diagonals (n+k+1,n-k) is a polynomial sequence of degree 2k. Assuming these observations to be true, the entire plane of rational values can be constructed from the single sequence R(0,p)! The values off the axes are simply extrapolated from values on and closer to the axes based on the polynomial form of the diagonals, with the proper sign. The sequence R(0,p) begins with 0, 1/2, 2-4/Pi, 17/2-24/Pi, 40-368/(3Pi) and twice the rational part of this sequence is A089165. The mathematica program given here is copied verbatim from the paper."
%C The first Mathematica code produces 0, 1, 4 - 8/Pi, 17 - 48/Pi, 80 - 736/(3*Pi), 401 - 3760/(3*Pi), 2084 - 98104/(15*Pi), 11073 - 521696/(15*Pi), 59712 - 19696256/(105*Pi), 325441 - 7156768/(7*Pi), 1788004 - 1769409304/(135*Pi); ... and the integer part gives the sequence.
%H Vincenzo Librandi, <a href="/A089165/b089165.txt">Table of n, a(n) for n = 0..200</a>
%H D. Atkinson and F. J. van Steenwijk, <a href="https://www.researchgate.net/publication/2834057_Infinite_Resistive_Lattices">Infinite Resistive Lattices</a>.
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%F G.f.: 1/((1-z)*sqrt(1-6*z+z^2)).
%F a(n) = Sum_{j=0..n} Sum_{i=0..2n} binomial(i, j)*binomial(j, i-j). - _Benoit Cloitre_, Oct 23 2004
%F a(n) = Sum_{k=0..n} C(n+k+1,2k+1)*A000984(k). - _Paul Barry_, Jun 03 2009
%F G.f.: d/dx atan(x*A006318(x)). - _Vladimir Kruchinin_
%F Recurrence: n*a(n) = (7*n-3)*a(n-1) - (7*n-4)*a(n-2) + (n-1)*a(n-3). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%p G:=1/((1-z)*sqrt(1-6*z+z^2)): Gser:=series(G,z=0,26): seq(coeff(Gser,z,n),n=0..23); # _Emeric Deutsch_, May 05 2006
%t alphas[beta_]:=Log[2-Cos[beta]+Sqrt[3+Cos[beta]*(Cos[beta]-4)]]; Rsqu[n_, p_]:=Simplify[(1/Pi)*Integrate[(1-Exp[ -Abs[n]*alphas[beta]]*Cos[p*beta])/Sinh[alphas[beta]], {beta, 0, Pi}]]; Table[Expand[2Rsqu[0, k]], {k, 0, 8}] (Evert-Jan D. Pol)
%t f[n_] := Sum[ Binomial[i, j] Binomial[j, i-j], {i, 0, 2n}, {j, 0, n}]; Table[ f@n, {n, 0, 21}] (* or *)
%t CoefficientList[ Series[ 1/((1 - x)Sqrt[1 - 6x + x^2]), {x, 0, 21}], x] (* _Robert G. Wilson v_, May 04 2006 *)
%o (PARI) a(n)=sum(i=0,2*n,sum(j=0,n,binomial(i,j)*binomial(j,i-j)))
%o (PARI) x + O(x^66); Vec(deriv(atan(x*(1-x-(1-6*x+x^2)^(1/2))/(2*x)))) \\ _Joerg Arndt_, Apr 21 2011
%Y Cf. A001850, A006318.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Dec 06 2003
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