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%I M3938 N1621 #180 Apr 19 2024 11:30:23
%S 0,1,5,25,129,681,3653,19825,108545,598417,3317445,18474633,103274625,
%T 579168825,3256957317,18359266785,103706427393,586889743905,
%U 3326741166725,18885056428537,107347191941249,610916200215241
%N a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).
%C From _Benoit Cloitre_, Jan 29 2002: (Start)
%C Array interpretation (first row and column are the natural numbers):
%C 1 2 3 ..j ... if b(i,j) = b(i-1,j) + b(i-1,j-1) + b(i,j-1) then a(n+1) = b(n,n)
%C 2 5 .........
%C .............
%C i........... b(i,j)
%C (End)
%C Number of ordered trees with 2n edges, having root of even degree, nonroot nodes of outdegree at most 2 and branches of odd length. - _Emeric Deutsch_, Aug 02 2002
%C Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - _Michael Somos_, Sep 24 2003
%C Number of peaks 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,0). Example: a(2)=5 because HH, HU*D, U*DH, UHD, U*DU*D, UU*DD contain 5 peaks (indicated by *). - _Emeric Deutsch_, Dec 06 2003
%C a(n) is the total number of HHs in all Schroeder (n+1)-paths. Example: a(2)=5 because UH*HD, H*H*H, UDH*H, H*HUD contain 5 HHs (indicated by *) and the other 18 Schroeder 3-paths contain no HHs. - _David Callan_, Jul 03 2006
%C a(n) is the total number of Hs in all Schroeder n-paths. Example: a(2)=5 as the Schroeder 2-paths are HH, DUH, DHU, HDU, DUDU and DDUU, and there are 5 H's. In general, a(n) is the total number of H..Hs (m+1 H's) in all Schroeder (n+m)-paths. - _FUNG Cheok Yin_, Jun 19 2021
%C a(n) is the number of points in Z^(n+1) that are L1 (Manhattan) distance <= n from the origin, or the number of points in Z^n that are L1 distance <= n+1 from the origin. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the (n+1)-dimensional cubic lattice as well as the (n+1)-st term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - _Shel Kaphan_, Dec 25 2022 [Edited by _Peter Munn_, Jan 05 2023]
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A002002/b002002.txt">Table of n, a(n) for n = 0..1000</a> (terms 0 through 100 were computed by Vincenzo Librandi)
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.
%H F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, <a href="http://arxiv.org/abs/1601.06690">Correlators for the Wigner-Smith time-delay matrix of chaotic cavities</a>, arXiv:1601.06690 [math-ph], 2016.
%H L. Ericksen, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.017">Lattice path combinatorics for multiple product identities</a>, J. of Statistical Planning and Inference 140 (2010) 2113-2226, see p. 2219.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 19.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H G. Rutledge and R. D. Douglass, <a href="http://www.jstor.org/stable/2301099">Integral functions associated with certain binomial coefficient sums</a>, Amer. Math. Monthly, 43 (1936), 27-32.
%F G.f.: ((1-x)/sqrt(1-6*x+x^2)-1)/2. - _Emeric Deutsch_, Aug 02 2002
%F E.g.f.: exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - _Vladeta Jovovic_, Mar 28 2004
%F a(n) = Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k+1). - _Paul Barry_, Sep 20 2004
%F a(n) = n * hypergeom([n + 1, -n + 1], [2], -1) = (n+1)*LegendreP(n+1,3) - (5*n+3)*LegendreP(n,3))/(2*n) for n > 0. - _Mark van Hoeij_, Jul 12 2010
%F G.f.: x*d/dx log(1/(1-x*A006318(x))). - _Vladimir Kruchinin_, Apr 19 2011
%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Aug 09 2011
%F G.f.: -1 + 1 / ( 1 - x / (1 - 4*x / (1 - x^2 / (1 - 4*x / (1 - x^2 / (1 - 4*x / ...)))))). - _Michael Somos_, Jan 03 2013
%F a(n) = Sum_{k=0..n} A201701(n,k)^2 = Sum_{k=0..n} A124182(n,k)^2 for n > 0. - _Philippe Deléham_, Dec 05 2011
%F D-finite with recurrence: 2*(6*n^2-12*n+5)*a(n-1)-(n-2)*(2*n-1)*a(n-2)-n*(2*n-3)*a(n)=0. - _Vaclav Kotesovec_, Oct 04 2012
%F a(n) ~ (3+2*sqrt(2))^n/(2^(5/4)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 04 2012
%F D-finite (an alternative): n*a(n) = (6-n)*a(n-6) + (14*n-72)*a(n-5) + (264-63*n)*a(n-4) + 100*(n-3)*a(n-3) + (114-63*n)*a(n-2) + 2*(7*n-6)*a(n-1), n >= 7. - _Fung Lam_, Feb 05 2014
%F a(n) = (-1)^(n-1)*Sum_{k=0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k) and n^3*a(n) = Sum_{k=0..n-1} (4*k^3+4*k^2+4*k+1)*binomial(n-1,k)*binomial(n+k,k). For each of the two equalities, both sides satisfy the same recurrence -- this follows from the Zeilberger algorithm. - _Zhi-Wei Sun_, Aug 30 2014
%F a(n) = hypergeom([1-n, -n], [1], 2) for n >= 1. - _Peter Luschny_, Nov 19 2014
%F Logarithmic derivative of A001003 (little Schroeder numbers). - _Paul D. Hanna_, May 17 2015
%F L.g.f.: L(x) = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1-x)^(-n) / n! = Sum_{n>=1} a(n)*x^n/n where exp(L(x)) = g.f. of A001003. - _Paul D. Hanna_, May 17 2015
%F a(n+1) = (1/2^(n+1)) * Sum_{k >= 0} (1/2^k) * binomial(n + k, n)*binomial(n + k, n + 1). - _Peter Bala_, Mar 02 2017
%F 2*a(n) = A110170(n), n > 0. - _R. J. Mathar_, Feb 10 2022
%F a(n) = (LegendreP(n,3) - LegendreP(n-1,3))/2. - _Mark van Hoeij_, Jul 14 2022
%F D-finite with recurrence n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - _R. J. Mathar_, Aug 01 2022
%F From _Peter Bala_, Nov 08 2022: (Start)
%F a(n) = (-1)^(n+1)*hypergeom( [n+1, -n+1], [1], 2) for n >= 1.
%F The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
%F From _Peter Bala_, Apr 18 2024: (Start)
%F G.f.: Sum_{n >= 1} binomial(2*n-1, n)*x^n/(1 - x)^(2*n) = x + 5*x^2 + 25*x^3 + 129*x^4 + ....
%F Row sums of A253283. (End)
%e G.f. = x + 5*x^2 + 25*x^3 + 129*x^4 + 681*x^5 + 3653*x^6 + 19825*x^7 + 108545*x^8 + ...
%p A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n,k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1,k); fi; end; seq(A064861(i,i+1),i=1..40);
%t CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}],x] (* _Harvey P. Dale_, Mar 17 2011 *)
%t a[ n_] := n Hypergeometric2F1[ n + 1, -n + 1, 2, -1] (* _Michael Somos_, Aug 09 2011 *)
%t a[ n_] := With[{m = Abs@n}, Sign[n] Sum[ Binomial[ m, k] Binomial[ m + k - 1, m], {k, m}]]; (* _Michael Somos_, Aug 09 2011 *)
%o (PARI) {a(n) = my(m = abs(n)); sign( n) * sum( k=0, m-1, binomial( m, k+1) * binomial( m+k, k))}; /* _Michael Somos_, Aug 09 2011 */
%o (Maxima) makelist(sum(binomial(n,k+1)*binomial(n+k,k), k, 0, n), n, 0, 21); \\ _Bruno Berselli_, May 19 2011
%o (Magma) [&+[Binomial(n,k+1)*Binomial(n+k,k): k in [0..n]]: n in [0..21]]; // _Bruno Berselli_, May 19 2011
%o (Sage)
%o a = lambda n: hypergeometric([1-n, -n], [1], 2) if n>0 else 0
%o [simplify(a(n)) for n in range(22)] # _Peter Luschny_, Nov 19 2014
%o (PARI) /* L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1-x)^(-n)/n! */
%o {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)/(1-x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, May 17 2015
%Y Bisection of A002003, Cf. A047781, A001003.
%Y a(n)=T(n, n+1), array T as in A050143.
%Y a(n)=T(n, n+1), array T as in A064861.
%Y Half the first differences of central Delannoy numbers (A001850).
%Y a(n)=T(n, n+1), array T as in A008288.
%Y Cf. A026002, A190666, A259554.
%K nonn,easy,changed
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _Clark Kimberling_
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