The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A026375 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k). 44
 1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605, 23973456915, 116312293305, 565280386625, 2751474553575, 13411044301945, 65448142561035, 319756851757695 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). 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 Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733. - Philippe Deléham, Oct 02 2007 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 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 Diagonal of rational function 1/(1 - (x^2 + 3*x*y + y^2)). - Gheorghe Coserea, Jul 29 2018 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi) Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5. D. Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4. Shu-Chiuan Chang, Robert Shrock, Structure of the Partition Function and transfer matrices for the Potts model in a magnetic field on lattice strips, J. Stat. Phys. 137 (2009) 667. D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From N. J. A. Sloane, May 11 2012 Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019. Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012. Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From N. J. A. Sloane, Sep 14 2012 J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019. H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From N. J. A. Sloane, Mar 16 2014 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7. FORMULA Representation by Gauss's hypergeometric function, in Maple notation: a(n)=hypergeom([ -n, 1/2 ], [ 1 ], -4). - Karol A. Penson, Apr 20 2001 This sequence is the binomial transform of A000984. - John W. Layman, Aug 11 2000; proved by Emeric Deutsch, Oct 26 2002 E.g.f.: exp(3*x)*I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 17 2002 G.f.: 1/sqrt(1-6*x+5*x^2). - Emeric Deutsch, Oct 26 2002 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 a(n) = [t^n](1+3*t+t^2)^n; a(n)=sum(j=ceiling(n/2)..n, 3^(2*j-n)*binomial(n, j)* binomial(j, n-j) ). - Emeric Deutsch, Jan 30 2004 a(n) = A026380(2*n-1) (n>0). - Emeric Deutsch, Feb 18 2004 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 a(n) = sum of squared coefficients of (1+x-x^2)^n - see triangle A084610. - Paul D. Hanna, Jul 18 2009 a(n) = sum of squares of coefficients of (1-x-x^2)^n. - Joerg Arndt, Jul 06 2011 a(n) = (1/Pi)*integral(x =-2..2, (3+x)^n/sqrt((2-x)*(2+x)). - Peter Luschny, Sep 12 2011 a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012 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 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 a(n) = GegenbauerC(n, -n, -3/2). - Peter Luschny, May 09 2016 a(n) = Sum_{k=0..n} 5^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019 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 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 EXAMPLE G.f. = 1 + 3*x + 11*x^2 + 45*x^3 + 195*x^4 + 873*x^5 + 3989*x^6 + ... MAPLE seq( add(binomial(n, k)*binomial(2*k, k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001 a := n -> simplify(GegenbauerC(n, -n, -3/2)): seq(a(n), n=0..22); # Peter Luschny, May 09 2016 MATHEMATICA Table[SeriesCoefficient[1/Sqrt[1-6*x+5*x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *) (* From Michael Somos, May 11 2014: (Start) *) a[ n_] := Sum[ Binomial[n, k] Binomial[2 k, k], {k, 0, n}]; a[ n_] := If[ n < 0, 0, Hypergeometric2F1[-n, 1/2, 1, -4]]; a[ n_] := If[ n < 0, 0, Coefficient[(1 + 3 x + x^2)^n, x, n]]; a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[Exp[3 x] BesselI[0, 2 x], {x, 0, n}]]; (* (End) *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + 3*x + x^2)^n, n))}; /* Michael Somos, Sep 09 2002 */ (Maxima) A026375(n):=coeff(expand((1+3*x+x^2)^n), x, n); makelist(A026375(n), n, 0, 12); /* Emanuele Munarini, Mar 02 2011 */ (PARI) a(n)={my(v=Vec((1-x-x^2)^n)); sum(k=1, #v, v[k]^2); } \\ Joerg Arndt, Jul 06 2011 (PARI) {a(n) = sum(k=0, n, 5^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019 (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 (Haskell) a026375 n = a026374 (2 * n) n  -- Reinhard Zumkeller, Feb 22 2014 (GAP) List([0..25], n->Sum([0..n], k->Binomial(n, k)*Binomial(2*k, k))); # Muniru A Asiru, Jul 29 2018 CROSSREFS Column 3 of A292627. Cf. A002893, A085362, A026380, A084610, A000172. First differences are in A085362. Bisection of A026380. 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). Sequence in context: A151125 A151126 A151127 * A151128 A049183 A049166 Adjacent sequences:  A026372 A026373 A026374 * A026376 A026377 A026378 KEYWORD nonn AUTHOR EXTENSIONS Definition simplified by N. J. A. Sloane, Feb 16 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 7 12:07 EDT 2020. Contains 333305 sequences. (Running on oeis4.)