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 A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k). (Formerly M3938 N1621) 29
 0, 1, 5, 25, 129, 681, 3653, 19825, 108545, 598417, 3317445, 18474633, 103274625, 579168825, 3256957317, 18359266785, 103706427393, 586889743905, 3326741166725, 18885056428537, 107347191941249, 610916200215241 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Benoit Cloitre, Jan 29 2002: (Start) Array interpretation (first row and column are the natural numbers): 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) 2 5 ......... ............. i........... b(i,j) (End) 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 Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - Michael Somos, Sep 24 2003 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 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 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 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] REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 through 100 were computed by Vincenzo Librandi) Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385. Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021. F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016. L. Ericksen, Lattice path combinatorics for multiple product identities, J. of Statistical Planning and Inference 140 (2010) 2113-2226, see p. 2219. Milan Janjic, Two Enumerative Functions Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019. M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32. FORMULA G.f.: ((1-x)/sqrt(1-6*x+x^2)-1)/2. - Emeric Deutsch, Aug 02 2002 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 a(n) = Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k+1). - Paul Barry, Sep 20 2004 a(n) = n * hypergeom([n + 1, -n + 1], , -1) = (n+1)*LegendreP(n+1,3) - (5*n+3)*LegendreP(n,3))/(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010 G.f.: x*d/dx log(1/(1-x*A006318(x))). - Vladimir Kruchinin, Apr 19 2011 a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 09 2011 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 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 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 a(n) ~ (3+2*sqrt(2))^n/(2^(5/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012 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 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 a(n) = hypergeom([1-n, -n], , 2) for n >= 1. - Peter Luschny, Nov 19 2014 Logarithmic derivative of A001003 (little Schroeder numbers). - Paul D. Hanna, May 17 2015 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 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 2*a(n) = A110170(n), n > 0. - R. J. Mathar, Feb 10 2022 a(n) = (LegendreP(n,3) - LegendreP(n-1,3))/2. - Mark van Hoeij, Jul 14 2022 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 From Peter Bala, Nov 08 2022: (Start) a(n) = (-1)^(n+1)*hypergeom( [n+1, -n+1], , 2) for n >= 1. 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) EXAMPLE G.f. = x + 5*x^2 + 25*x^3 + 129*x^4 + 681*x^5 + 3653*x^6 + 19825*x^7 + 108545*x^8 + ... MAPLE 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); MATHEMATICA CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x, 0, 30}], x] (* Harvey P. Dale, Mar 17 2011 *) a[ n_] := n Hypergeometric2F1[ n + 1, -n + 1, 2, -1] (* Michael Somos, Aug 09 2011 *) a[ n_] := With[{m = Abs@n}, Sign[n] Sum[ Binomial[ m, k] Binomial[ m + k - 1, m], {k, m}]]; (* Michael Somos, Aug 09 2011 *) PROG (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 */ (Maxima) makelist(sum(binomial(n, k+1)*binomial(n+k, k), k, 0, n), n, 0, 21); \\ Bruno Berselli, May 19 2011 (Magma) [&+[Binomial(n, k+1)*Binomial(n+k, k): k in [0..n]]: n in [0..21]]; // Bruno Berselli, May 19 2011 (Sage) a = lambda n: hypergeometric([1-n, -n], , 2) if n>0 else 0 [simplify(a(n)) for n in range(22)] # Peter Luschny, Nov 19 2014 (PARI) /* L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1-x)^(-n)/n! */ {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {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)} for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, May 17 2015 CROSSREFS Bisection of A002003, Cf. A047781, A001003. a(n)=T(n, n+1), array T as in A050143. a(n)=T(n, n+1), array T as in A064861. Half the first differences of central Delannoy numbers (A001850). a(n)=T(n, n+1), array T as in A008288. Cf. A026002, A190666, A259554. Sequence in context: A270767 A026718 A060928 * A182626 A351050 A184139 Adjacent sequences: A001999 A002000 A002001 * A002003 A002004 A002005 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Simon Plouffe EXTENSIONS More terms from Clark Kimberling STATUS approved

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Last modified December 4 21:08 EST 2023. Contains 367565 sequences. (Running on oeis4.)