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A002002 Sum(k=0..n-1, binomial(n,k+1) * binomial(n+k,k) ).
(Formerly M3938 N1621)
17
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) = 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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

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

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 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-z)/sqrt(1-6*z+z^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, k)*binomial(n+k+1, k+1)). - Paul Barry, Sep 20 2004

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

G.f. x*d/dx log(1/(1-x*A006318(x))). - Vladimir Kruchinin, Apr 19 2011

a(-n) = -a(n). - 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<=k<=n} A201701(n,k)^2 = Sum_{k, 0<=k<=n} A124182(n,k)^2 for n>0 . - Philippe Deléham, Dec 05 2011

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

Recurrence (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

EXAMPLE

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) = local(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

CROSSREFS

Cf. A002003, A047781.

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.

Sequence in context: A026718 A060928 * A182626 A184139 A102893 A094602

Adjacent sequences:  A001999 A002000 A002001 * A002003 A002004 A002005

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Clark Kimberling

STATUS

approved

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Last modified October 31 10:52 EDT 2014. Contains 248861 sequences.