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%I #137 Mar 22 2023 22:00:08
%S 0,1,4,19,96,501,2668,14407,78592,432073,2390004,13286043,74160672,
%T 415382397,2333445468,13141557519,74174404608,419472490257,
%U 2376287945572,13482186743203,76598310928096,435730007006341,2481447593848524,14146164790774359
%N a(n) = Sum_{k=0..n-1} binomial(n-1,k)*binomial(n+k,k). Also a(n) = T(n,n), array T as in A049600.
%C Also main diagonal of array: m(i,1)=1, m(1,j)=j, m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 3 4 ... / 1 4 9 16 ... / 1 6 19 44 ... / 1 8 33 96 ... /. - _Benoit Cloitre_, Aug 05 2002
%C This array is now listed as A142978, where some conjectural congruences for the present sequence are given. - _Peter Bala_, Nov 13 2008
%C Convolution of central Delannoy numbers A001850 and little Schroeder numbers A001003. Hankel transform is 2^C(n+1,2)*A007052(n). - _Paul Barry_, Oct 07 2009
%C Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0 <= r <= n and the other terms recursively with T(r,c) = T(r-1,c-1) + 2*T(r,c-1). The sum of the last terms in the rows is Sum_{r=0..n} T(r,r) = a(n+1). Example: For n=4 the triangle has the rows 1; 4 9; 6 16 41; 4 14 44 129; 1 6 26 96 321 having sum of last terms 1 + 9 + 41 + 129 + 321 = 501 = a(5). - _J. M. Bergot_, Feb 15 2013
%C a(n) = A049600(2*n,n), when A049600 is seen as a triangle read by rows. - _Reinhard Zumkeller_, Apr 15 2014
%C a(n-1) for n > 1 is the number of assembly trees with the connected gluing rule for cycle graphs with n vertices. - _Nick Mayers_, Aug 16 2018
%H Seiichi Manyama, <a href="/A047781/b047781.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)
%H A. Bacher, <a href="http://arxiv.org/abs/1301.1365">Directed and multi-directed animals on the square lattice with next nearest neighbor edges</a>, arXiv preprint arXiv:1301.1365 [math.CO], 2013. See D(t). - From _N. J. A. Sloane_, Feb 14 2013
%H C. Banderier and P. Hitczenko, <a href="http://doi.org/10.1016/j.dam.2011.12.011">Enumeration and asymptotics of restricted compositions having the same number of parts</a>, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Table 2.
%H M. Bona and A. Vince, <a href="https://arxiv.org/abs/1204.3842">The Number of Ways to Assemble a Graph</a>, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
%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 A. Dougherty, N. Mayers, and R. Short, <a href="https://arxiv.org/abs/1807.08079"> How to Build a Graph in n Days: Some Variants on Graph Assembly</a>, arXiv preprint arXiv:1807.08079 [math.CO], 2018.
%H Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.
%H Steffen Eger, <a href="https://arxiv.org/abs/1704.04964">The Combinatorics of Weighted Vector Compositions</a>, arXiv:1704.04964 [math.CO], 2017.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%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 D-finite with recurrence n*(2*n-3)*a(n) - (12*n^2-24*n+8)*a(n-1) + (2*n-1)*(n-2)*a(n-2) = 0. - _Vladeta Jovovic_, Aug 29 2004
%F a(n+1) = Sum_{k=0..n} binomial(n, k)*binomial(n+1, k+1)*2^k. - _Paul Barry_, Sep 20 2004
%F a(n) = Sum_{k=0..n} T(n, k), array T as in A008288.
%F If shifted one place left, the third binomial transform of A098660. - _Paul Barry_, Sep 20 2004
%F G.f.: ((1+x)/sqrt(1-6x+x^2)-1)/4. - _Paul Barry_, Sep 20 2004, simplified by _M. F. Hasler_, Oct 09 2012
%F E.g.f. for sequence shifted left: Sum_{n>=0} a(n+1)*x^n/n! = exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+BesselI(1, 2*sqrt(2)*x)/sqrt(2)). - _Paul Barry_, Sep 20 2004
%F a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)*2^(n-k-1); a(n+1) = 2^n*Hypergeometric2F1(-n,-n-1;1;1/2). - _Paul Barry_, Feb 08 2011
%F a(n) ~ 2^(1/4)*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 08 2012
%F Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n >= 7. - _Fung Lam_, Feb 05 2014
%F a(n) = A241023(n) / 4. - _Reinhard Zumkeller_, Apr 15 2014
%F a(n) = Hyper2F1([-n, n], [1], -1)/2 for n > 0. - _Peter Luschny_, Aug 02 2014
%F n^2*a(n) = Sum_{k=0..n-1} (2*k^2+2*k+1)*binomial(n-1,k)*binomial(n+k,k). By the Zeilberger algorithm, both sides of the equality satisfy the same recurrence. - _Zhi-Wei Sun_, Aug 30 2014
%F a(n) = [x^n] (1/2) * ((1+x)/(1-x))^n for n > 0. - _Seiichi Manyama_, Jun 07 2018
%p a := proc(n) local k; add(binomial(n-1,k)*binomial(n+k,k),k=0..n-1); end;
%t Table[SeriesCoefficient[x*((1+x)-Sqrt[1-6*x+x^2])/(4*x*Sqrt[1-6*x+x^2]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%t a[n_] := Hypergeometric2F1[1-n, n+1, 1, -1]; a[0] = 0; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Feb 26 2013 *)
%t a[n_] := Sum[ Binomial[n - 1, k] Binomial[n + k, k], {k, 0, n - 1}]; Array[a, 25] (* _Robert G. Wilson v_, Aug 08 2018 *)
%o (Maxima) makelist(if n=0 then 0 else sum(binomial(n-1, k)*binomial(n+k, k), k, 0, n-1), n, 0, 22); \\ _Bruno Berselli_, May 19 2011
%o (Magma) [n eq 0 select 0 else &+[Binomial(n-1, k)*Binomial(n+k, k): k in [0..n-1]]: n in [0..22]]; // _Bruno Berselli_, May 19 2011
%o (PARI) A047781(n)=polcoeff((1+x)/sqrt(1+(O(x^n)-6)*x+x^2),n)\4 \\ _M. F. Hasler_, Oct 09 2012
%o (Haskell)
%o a047781 n = a049600 (2 * n) n -- _Reinhard Zumkeller_, Apr 15 2014
%o (Python)
%o from sympy import binomial
%o def a(n):
%o return sum(binomial(n - 1, k) * binomial(n + k, k) for k in range(n))
%o print([a(n) for n in range(51)]) # _Indranil Ghosh_, Apr 18 2017
%o (Python)
%o from math import comb
%o def A047781(n): return sum(comb(n,k)**2*k<<k-1 for k in range(1,n+1))//n if n else 0 # _Chai Wah Wu_, Mar 22 2023
%Y Cf. A002003. Column 1 of A296129.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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