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 A026002 a(n) = T(n,n+2), where T = Delannoy triangle (A008288). 6
 1, 7, 41, 231, 1289, 7183, 40081, 224143, 1256465, 7059735, 39753273, 224298231, 1267854873, 7178461215, 40704778785, 231128079903, 1314016698401, 7478998203943, 42612705597769, 243025194476551, 1387226559025961, 7924982285747247 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of U steps in all lattice paths from (0,0) to (2n,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis (i.e., Schroeder paths). For example, a(2)=7, counting the U's in HH, UDUD, UUDD, UHD, HUD and UDH. - Emeric Deutsch, Dec 06 2003 Number of UH's in all lattice paths from (0,0) to (2n+2,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis (i.e., Schroeder paths). For example, a(2)=7, counting the UH's, shown between parentheses, in the 22 (=A006318(3)) Schroeder paths of length 6: HHH, HHUD, HUDH, HUDUD, H(UH)D, HUUDD, (UH)DH, (UH)DUD, UUDDH, UUDDUD, (UH)HD, (UH)UDD, UUDHD, UUDUDD, U(UH)DD, UUUDDD, UDHH, UDHUD, UDUDH, UDUDUD, UD(UH)D and UDUUDD. - Emeric Deutsch, Jul 16 2005 Number of walks from (0,0) to (n+2,n) using steps from {E,N,NE}. - Shanzhen Gao, May 25 2011 Conjecture: define an infinite array to have m(n,1) = m(1,n) = n*(n-1)+1 in the first row and column, and m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j); then m(n,n) = a(n). - J. M. Bergot, Apr 24 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..200 L. Ferrari, E. Munarini, Enumeration of Edges in Some Lattices of Paths, J. Int. Seq. 17 (2014) #14.1.5. FORMULA From Emeric Deutsch, Dec 06 2003: (Start) a(n) = (1/n)*Sum_{k=0..n} k*binomial(n, k)*binomial(n+k, k+1). G.f.: 1/2 - 1/(2*z) + (1-4*z+z^2)/(2*z*sqrt(1-6*z+z^2)). a(n) = Sum_{k=0..floor(n/2)} k*A110220(n, k). - Emeric Deutsch, Jul 16 2005 a(n) = Sum_{k=0..n} C(n, k)*C(n+2, k)*2^k. - Paul Barry, Jan 23 2006 a(n) = Jacobi_P(n, 2, 0, 3). - Paul Barry, Jan 23 2006 a(n) = (-1)^n*((2*n-1)*LegendreP(n,-3)-LegendreP(n-1,-3))/(2*n+2). - Mark van Hoeij, Oct 31 2011 Recurrence: (n+1)*(6*n-7)*a(n) = (36*n^2-23*n+7)*a(n-1) - (6*n^2-n-21)*a(n-2) + (n-3)*a(n-3). - Vaclav Kotesovec, Oct 08 2012 a(n) ~ sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012 a(n) = hypergeom([-n-1, -n+1], [1], 2). - Peter Luschny, Nov 19 2014 From Peter Bala, Mar 02 2017: (Start) a(n+1) = 1/2^(n+1) * Sum_{k >= 2} 1/2^k * binomial(n + k, n)*binomial(n + k, n + 2). (n+1)*(n-1)^2*a(n) = (2*n-1)*(3*n^2 -3*n +1)*a(n-1) - (n-2)*n^2*a(n-2) with a(1) = 1 and a(2) = 7. (End) a(n) = A001850(n) - A006318(n). - Matthew Niemiro, Jan 31 2020 MAPLE a:=n->(1/n)*sum(k*binomial(n, k)*binomial(n+k, k+1), k=0..n): seq(a(n), n=1..25); # Emeric Deutsch MATHEMATICA Table[SeriesCoefficient[1/2-1/(2*x)+(1-4*x+x^2)/(2*x*Sqrt[1-6*x+x^2]), {x, 0, n}], {n, 1, 25}] (* Vaclav Kotesovec, Oct 08 2012 *) PROG (PARI) my(x='x+O('x^66)); Vec( 1/2-1/(2*x)+(1-4*x+x^2)/(2*x*sqrt(1-6*x+x^2)) ) \\ Joerg Arndt, May 04 2013 (Sage) a = lambda n: hypergeometric([-n-1, -n+1], [1], 2) [simplify(a(n)) for n in (1..25)] # Peter Luschny, Nov 19 2014 (MAGMA) [(1/n)*(&+[k*Binomial(n, k)*Binomial(n+k, k+1): k in [0..n]]): n in [1..25]]; // G. C. Greubel, Feb 13 2020 (GAP) List([1..25], n-> (1/n)*Sum([0..n], k-> k*Binomial(n, k)*Binomial(n+k, k+1) )); # G. C. Greubel, Feb 13 2020 CROSSREFS Cf. A002002, A008288, A110220, A190666. Sequence in context: A144635 A097165 A152268 * A173409 A057009 A140480 Adjacent sequences:  A025999 A026000 A026001 * A026003 A026004 A026005 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified November 24 01:40 EST 2020. Contains 338603 sequences. (Running on oeis4.)