A026375 - OEIS

%I #178 Aug 06 2024 04:32:50

%S 1,3,11,45,195,873,3989,18483,86515,408105,1936881,9238023,44241261,

%T 212601015,1024642875,4950790605,23973456915,116312293305,

%U 565280386625,2751474553575,13411044301945,65448142561035,319756851757695

%N a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).

%C a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=0; also a(n)=T(2n,n).

%C Partial sums of A085362. Number of bilateral Schroeder paths (i.e., lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at odd (positive or negative) levels. Example: a(2)=11 because we have HUD, UDH, UDUD, UUDD, UDDU, their reflections in the x-axis and HH. - _Emeric Deutsch_, Jan 30 2004

%C Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733. - _Philippe Deléham_, Oct 02 2007

%C Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in three colors. - _N-E. Fahssi_, Feb 05 2008

%C Equals INVERT transform of A109033: (1, 2, 6, 22, 88, ...), INVERTi transform of A111966, binomial transform of A000984, and inverse Binomial transform of A081671. Convolved with A002212: (1, 3, 10, 36, ...) = A026376: (1, 6, 30, 144, ...). Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906, ...). - _Gary W. Adamson_, May 17 2009

%C Diagonal of array with rational generating function 1/(1 - (x^2 + 3*x*y + y^2)). - _Gheorghe Coserea_, Jul 29 2018

%C a(n) == 0 (mod 3) if and only if n is in A081606. - _Fabio Visonà_, Aug 03 2023

%H Seiichi Manyama, <a href="/A026375/b026375.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)

%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

%H David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Callan2/callan204.html">A combinatorial interpretation for an identity of Barrucand</a>, JIS 11 (2008) 08.3.4.

%H Shu-Chiuan Chang and Robert Shrock, <a href="https://doi.org/10.1007/s10955-009-9868-0">Structure of the Partition Function and transfer matrices for the Potts model in a magnetic field on lattice strips</a>, J. Stat. Phys. 137 (2009) 667.

%H D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="https://doi.org/10.37236/2034">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012

%H Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

%H Rui Duarte and António Guedes de Oliveira, <a href="https://www.cmup.pt/sites/default/files/2023-08/GF_LP_corrected_0.pdf">Generating functions of lattice paths</a>, Univ. do Porto (Portugal 2023).

%H Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="https://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012.

%H Francesc Fite and Andrew V. Sutherland, <a href="https://arxiv.org/abs/1203.1476">Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1</a>, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012

%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/a/4746447/573047">Proof that a(n) == 0 (mod 3) if and only if n is in A081606</a>.

%H László Németh, <a href="https://arxiv.org/abs/1905.13475">Tetrahedron trinomial coefficient transform</a>, arXiv:1905.13475 [math.CO], 2019.

%H H. D. Nguyen and D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014

%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

%F Representation by Gauss's hypergeometric function, in Maple notation: a(n)=hypergeom([ -n, 1/2 ], [ 1 ], -4). - _Karol A. Penson_, Apr 20 2001

%F This sequence is the binomial transform of A000984. - _John W. Layman_, Aug 11 2000; proved by _Emeric Deutsch_, Oct 26 2002

%F E.g.f.: exp(3*x)*I_0(2x), where I_0 is Bessel function. - _Michael Somos_, Sep 17 2002

%F G.f.: 1/sqrt(1-6*x+5*x^2). - _Emeric Deutsch_, Oct 26 2002

%F D-finite with recurrence: n*a(n)-3*(2*n-1)*a(n-1)+5*(n-1)*a(n-2)=0 for n > 1. - _Emeric Deutsch_, Jan 24 2004

%F From _Emeric Deutsch_, Jan 30 2004: (Start)

%F a(n) = [t^n](1+3*t+t^2)^n;

%F a(n) = Sum_{j=ceiling(n/2)..n} 3^(2*j-n)*binomial(n, j)*binomial(j, n-j). (End)

%F a(n) = A026380(2*n-1) (n>0). - _Emeric Deutsch_, Feb 18 2004

%F G.f.: 1/(1-x-2*x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x... (continued fraction). - _Paul Barry_, Jan 06 2009

%F a(n) = sum of squared coefficients of (1+x-x^2)^n - see triangle A084610. - _Paul D. Hanna_, Jul 18 2009

%F a(n) = sum of squares of coefficients of (1-x-x^2)^n. - _Joerg Arndt_, Jul 06 2011

%F a(n) = (1/Pi)*Integral_{x=-2..2} ((3+x)^n/sqrt((2-x)*(2+x))) dx. - _Peter Luschny_, Sep 12 2011

%F a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 08 2012

%F G.f.: G(0)/(1-x), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 24 2013

%F 0 = a(n)*(+25*a(n+1) - 45*a(n+2) + 10*a(n+3)) + a(n+1)*(-15*a(n+1) + 36*a(n+2) - 9*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, May 11 2014

%F a(n) = GegenbauerC(n, -n, -3/2). - _Peter Luschny_, May 09 2016

%F a(n) = Sum_{k=0..n} 5^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - _Seiichi Manyama_, Apr 22 2019

%F a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - _Seiichi Manyama_, May 04 2019

%F a(n) = (1/Pi) * Integral_{x = -1..1} (1 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (5 - 4*x^2)^n/sqrt(1 - x^2) dx. - _Peter Bala_, Jan 27 2020

%F From _Peter Bala_, Jan 10 2022: (Start)

%F 1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + ... is the o.g.f. of A002212.

%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

%e G.f. = 1 + 3*x + 11*x^2 + 45*x^3 + 195*x^4 + 873*x^5 + 3989*x^6 + ...

%p seq( add(binomial(n,k)*binomial(2*k,k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

%p a := n -> simplify(GegenbauerC(n, -n, -3/2)):

%p seq(a(n), n=0..22); # _Peter Luschny_, May 09 2016

%t Table[SeriesCoefficient[1/Sqrt[1-6*x+5*x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)

%t (* From _Michael Somos_, May 11 2014: (Start) *)

%t a[ n_] := Sum[ Binomial[n, k] Binomial[2 k, k], {k, 0, n}];

%t a[ n_] := If[ n < 0, 0, Hypergeometric2F1[-n, 1/2, 1, -4]];

%t a[ n_] := If[ n < 0, 0, Coefficient[(1 + 3 x + x^2)^n, x, n]];

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[Exp[3 x] BesselI[0,2 x], {x, 0, n}]];

%t (* (End) *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + 3*x + x^2)^n, n))}; /* _Michael Somos_, Sep 09 2002 */

%o (Maxima) A026375(n):=coeff(expand((1+3*x+x^2)^n),x,n);

%o makelist(A026375(n),n,0,12); /* _Emanuele Munarini_, Mar 02 2011 */

%o (PARI) a(n)={my(v=Vec((1-x-x^2)^n)); sum(k=1,#v, v[k]^2);} \\ _Joerg Arndt_, Jul 06 2011

%o (PARI) {a(n) = sum(k=0, n, 5^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ _Seiichi Manyama_, Apr 22 2019

%o (PARI) {a(n) = sum(k=0, n\2, 3^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ _Seiichi Manyama_, May 04 2019

%o (Haskell)

%o a026375 n = a026374 (2 * n) n  -- _Reinhard Zumkeller_, Feb 22 2014

%o (GAP) List([0..25],n->Sum([0..n],k->Binomial(n,k)*Binomial(2*k,k))); # _Muniru A Asiru_, Jul 29 2018

%Y Column 3 of A292627. Column 1 of A110165. Central column of A272866.

%Y Cf. A002893, A084610, A000172, A002212.

%Y First differences are in A085362. Bisection of A026380.

%Y m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

%K nonn

%O 0,2

%A _Clark Kimberling_

%E Definition simplified by _N. J. A. Sloane_, Feb 16 2012