A292627 -id:A292627 - cp's OEIS frontend

A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605, 23973456915, 116312293305, 565280386625, 2751474553575, 13411044301945, 65448142561035, 319756851757695

Offset: 0

Clark Kimberling

nonn

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 array with rational generating function 1/(1 - (x^2 + 3*x*y + y^2)). - Gheorghe Coserea, Jul 29 2018
a(n) == 0 (mod 3) if and only if n is in A081606. - Fabio Visonà, Aug 03 2023

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

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
Shu-Chiuan Chang and 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, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
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.
Mathematics Stack Exchange, Proof that a(n) == 0 (mod 3) if and only if n is in A081606.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
H. D. Nguyen and 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.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Column 3 of A292627. Column 1 of A110165. Central column of A272866.
Cf. A002893, A084610, A000172, A002212.
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).

List([0..25],n->Sum([0..n],k->Binomial(n,k)*Binomial(2*k,k))); # Muniru A Asiru, Jul 29 2018

a026375 n = a026374 (2 * n) n  -- Reinhard Zumkeller, Feb 22 2014

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

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) *)

A026375(n):=coeff(expand((1+3*x+x^2)^n),x,n);
makelist(A026375(n),n,0,12); /* Emanuele Munarini, Mar 02 2011 */

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x + x^2)^n, n))}; /* Michael Somos, Sep 09 2002 */

a(n)={my(v=Vec((1-x-x^2)^n)); sum(k=1,#v, v[k]^2);} \\ Joerg Arndt, Jul 06 2011

{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, n\2, 3^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

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
From Emeric Deutsch, Jan 30 2004: (Start)
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). (End)
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))) dx. - 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
From Peter Bala, Jan 10 2022: (Start)
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.
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)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

Definition simplified by N. J. A. Sloane, Feb 16 2012

A081671 Expansion of e.g.f. exp(4x) * I_0(2x).

Original entry on oeis.org

1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, 13233628, 75682512, 434662684, 2505229744, 14482673832, 83940771168, 487610895942, 2838118247064, 16547996212044, 96635257790352, 565107853947444, 3308820294176016, 19395905063796312, 113814537122646432

Offset: 0

Paul Barry, Mar 28 2003

Binomial transform of A026375. Second binomial transform of A000984.
Largest coefficient of (1+4x+x^2)^n. - Paul Barry, Dec 15 2003
Row sums of triangle in A124574. - Philippe Deléham, Sep 27 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 4 colors. - N-E. Fahssi, Feb 05 2008
Diagonal of rational function 1/(1 - (x^2 + 4*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Isaac DeJager, Madeleine Naquin and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
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.
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
Rigoberto Flórez, Leandro Junes and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Column 4 of A292627.

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).

seq(simplify(2^n*hypergeom([-n,1/2], [1], -2)),n=0..23); # Peter Luschny, Apr 26 2016
seq(simplify(GegenbauerC(n,-n,-2)),n=0..23); # Peter Luschny, May 09 2016

Table[SeriesCoefficient[1/Sqrt[(1-2*x)*(1-6*x)],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a(n):=coeff(expand((1+4*x+x^2)^n),x^n);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

x='x+O('x^66); Vec(1/sqrt((1-2*x)*(1-6*x))) \\ Joerg Arndt, May 07 2013

{a(n) = sum(k=0, n\2, 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

a(n) = Sum_{m=0..n} Sum_{k=0..m} C(n, m)*C(m, k)*C(2k, k).
G.f.: 1/sqrt((1-2*x)*(1-6*x)). - Vladeta Jovovic, Oct 09 2003
a(n) = Sum_{k=0..n} 2^(n-k) * C(n, k) * C(2*k, k). - Paul Barry, Jan 27 2005
a(n) = Sum_{k=0..n} 6^(n-k) * (-1)^k * C(n,k) * C(2*k,k). - Paul D. Hanna, Dec 09 2018
D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 12*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(3/(2*Pi*n))*6^n. - Vaclav Kotesovec, Oct 13 2012
a(n) = 2^n*hypergeom([-n,1/2], [1], -2). - Peter Luschny, Apr 26 2016
a(n) = GegenbauerC(n, -n, -2). - Peter Luschny, May 09 2016
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
a(n) = (1/Pi) * Integral_{x = -1..1} (2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (6 - 4*x^2)^n/sqrt(1 - x^2) dx . - Peter Bala, Jan 27 2020
From Peter Bala, Jan 10 2022: (Start)
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 4*x + 17*x^2 + 76*x^3 + 354*x^4 + ... is the o.g.f. of A005572.
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)
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A098409 Expansion of 1/(sqrt(1-3x)sqrt(1-7*x)).

Original entry on oeis.org

1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, 10153775, 67313377, 448691985, 3004182349, 20188647185, 136094684907, 919884469275, 6232016686995, 42305974804575, 287706424085745, 1959685788407025, 13367193276457881, 91295551930615005, 624255065007468207

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A081671. 3rd binomial transform of A000984. Binomial transform is A098410.
Largest coefficient of (1+5*x+x^2)^n; row sums of triangle in A126331. - 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 five colors. - N-E. Fahssi, Feb 05 2008
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 can have five colors. - N-E. Fahssi, Mar 31 2008
Diagonal of rational function 1/(1 - (x^2 + 5*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Isaac DeJager, Madeleine Naquin and 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 (the sequence b_{5,n}).
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column 5 of A292627. Cf. A182401.

Table[SeriesCoefficient[1/(Sqrt[1-3*x]*Sqrt[1-7*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
CoefficientList[Series[1/(Sqrt[1-3x] Sqrt[1-7x]),{x,0,30}],x] (* Harvey P. Dale, Jun 20 2015 *)

a(n):=coeff(expand((1+5*x+x^2)^n),x^n);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

x='x+O('x^66); Vec(1/(sqrt(1-3*x)*sqrt(1-7*x))) \\ Joerg Arndt, May 11 2013

{a(n) = sum(k=0, n, 7^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 5^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

G.f.: 1/sqrt(1-10*x+21*x^2).
E.g.f.: exp(5x)*BesselI(0, 2x).
a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n,k)*binomial(2k,k). - Paul Barry, Mar 08 2005
a(n) = [x^n] (1+5*x+x^2)^n. - Emanuele Munarini, Apr 27 2012
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ 7^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012
a(n) = Sum_{k=0..n} 7^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
From Peter Bala, Jan 10 2022: (Start)
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 5*x + 26*x^2 + 140*x^3 + 777*x^4 + ... is the o.g.f. of A182401.
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.
a(n) = (1/Pi) * Integral_{x = -1..1} (3 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (7 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A098410 Expansion of 1/(sqrt(1-4x)sqrt(1-8*x)).

Original entry on oeis.org

1, 6, 38, 252, 1734, 12276, 88796, 652728, 4856902, 36478404, 275975028, 2099978568, 16054486044, 123213933576, 948713646072, 7325088811632, 56692748053062, 439689331938276, 3416328042565124, 26587566855421608, 207218159714453044, 1617124976299315224, 12634892752595949192

Offset: 0

Paul Barry, Sep 07 2004

Convolution of A000984(n) and 2^n*A000984(n). Convolution of A000984(n) and A059304. 4th binomial transform of A000984.
Largest coefficient of (1 + 6*x + x^2)^n. - 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 can have 6 colors. - N-E. Fahssi, Mar 31 2008
Self-convolution of a(n)/4^n gives A126646. - Vladimir Reshetnikov, Oct 10 2016
Diagonal of rational function 1/(1 - (x^2 + 6*x*y + y^2)). - Gheorghe Coserea, Aug 03 2018

			G.f. = 1 + 6*x + 38*x^2 + 252*x^3 + 1734*x^4 + 12276*x^5 + 88796*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
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 (the sequence b_{6,n}).
Rigoberto Flórez, Leandro Junes and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column 6 of A292627. Cf. A025230, A104454 (binomial transf.)

Table[SeriesCoefficient[1/(Sqrt[1-4*x]*Sqrt[1-8*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
a[ n_] := If[n < 0, 0, 4^n Hypergeometric2F1[-n, 1/2, 1, -1]]; (* Michael Somos, May 06 2017 *)
a[ n_] := SeriesCoefficient[ D[ InverseJacobiSD[2 x, -1] / 2, x], {x, 0, 2 n}]; (* Michael Somos, May 06 2017 *)

x='x+O('x^66); Vec(1/sqrt(1-12*x+32*x^2)) \\ Joerg Arndt, May 11 2013

{a(n) = sum(k=0, n, 8^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 6^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

a = lambda n: 4^n*hypergeometric([-n, 1/2], [1], -1)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

G.f.: 1/sqrt(1 - 12*x + 32*x^2).
E.g.f.: exp(6*x)*BesselI(0, 2*x).
a(n) = Sum_{k=0..n} 2^k*binomial(2*k, k)*binomial(2*(n-k), n-k).
a(n) = Sum_{k=0..n} 4^(n-k)*binomial(n,k)*binomial(2k,k). - Paul Barry, Mar 08 2005
D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) - 32*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ 2^(3*n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 15 2012
a(n) = 4^n*hypergeometric([-n, 1/2], [1], -1). - Peter Luschny, May 19 2015
a(n) = Sum_{k=0..n} 8^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 6^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
From Peter Bala, Jan 10 2022: (Start)
3*x + x^2*exp(Sum_{n >= 1} a(n)*x^n/n) = 3*x + x^2 + 6*x^3 + 37*x^4 + 234*x^5 + 1514*x^6 + ... is the o.g.f. of A025230.
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.
a(n) = (1/Pi) * Integral_{x = -1..1} (4 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (8 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)

A186925 Coefficient of x^n in (1+n*x+x^2)^n.

Original entry on oeis.org

1, 1, 6, 45, 454, 5775, 88796, 1602447, 33213510, 777665691, 20302315252, 584774029983, 18422140045596, 630132567760345, 23257790717110392, 921362075184792825, 38994274473840538182, 1755943506127367745795, 83829045032101462204100, 4229207755493569286374167

Offset: 0

Emanuele Munarini, Mar 02 2011

Seiichi Manyama, Table of n, a(n) for n = 0..386 (terms 0..100 from Vincenzo Librandi)

Main diagonal of A292627.

Cf. A092366, A187018, A187019, A187021, A070910, A292629.

P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 02 2011

Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k), {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[GegenbauerC[n, -n, -n/2] + KroneckerDelta[n, 0], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)
Table[SeriesCoefficient[(1 + n*x + x^2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 13 2023 *)

a(n):=coeff(expand((1+n*x+x^2)^n),x,n);

makelist(ultraspherical(n,-n,-n/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */
makelist(a(n),n,0,20);

{a(n) = sum(k=0, n, (n-2)^(n-k)*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, May 01 2019

a(n) = polcoef((1+n*x+x^2)^n, n); \\ Michel Marcus, May 01 2019

a(n) = [x^n] (1+n*x+x^2)^n.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^(n-2*k).
a(n) ~ BesselI(0,2) * n^n. - Vaclav Kotesovec, Apr 17 2014
a(n) = GegenbauerPoly(n,-n,-n/2). - Emanuele Munarini, Oct 20 2016
From Ilya Gutkovskiy, Sep 20 2017: (Start)
a(n) = [x^n] 1/sqrt((1 + 2*x - n*x)*(1 - 2*x - n*x)).
a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*x). (End)
From Seiichi Manyama, May 01 2019: (Start)
a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (n+2)^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). (End)
a(n) = (1/4)^n * Sum_{k=0..n} (n-2)^k * (n+2)^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A104454 Expansion of 1/(sqrt(1-5x)*sqrt(1-9x)).

Original entry on oeis.org

1, 7, 51, 385, 2995, 23877, 194109, 1602447, 13389075, 112935445, 959783881, 8206116387, 70507643101, 608271899515, 5265458413875, 45711784088145, 397829544860115, 3469772959954245, 30319709631711225, 265383615634224675, 2326318766651511945, 20419439617056272415

Offset: 0

Paul Barry, Mar 08 2005

Fifth binomial transform of A000984. In general, the k-th binomial transform of A000984 will have g.f. 1/(sqrt(1-k*x)*sqrt(1-(k+4)*x)) and a(n)=sum{i=0..n, C(n,i)C(2i,i)k^(n-i)}.

Diagonal of rational function 1/(1 - (x^2 + 7*x*y + y^2)). - Gheorghe Coserea, Aug 03 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column 7 of A292627.

Cf. A081671, A098409, A098410.

CoefficientList[Series[1/(Sqrt[1-5x] Sqrt[1-9x]),{x,0,30}],x] (* Harvey P. Dale, Apr 11 2012 *)

x='x+O('x^66); Vec(1/sqrt(1-14*x+45*x^2)) \\ Joerg Arndt, May 13 2013

{a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 7^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

G.f.: 1/sqrt(1-14*x+45*x^2).
E.g.f.: exp(7x)*BesselI(0, 2x)
a(n) = Sum_{k=0..n} 5^(n-k)*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 7*(2*n-1)*a(n-1) - 45*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n+1)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012
a(n) = Sum_{k=0..n} 9^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 7^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * 9^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * binomial(2*j,j).
D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A307847 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k^2)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 7, 1, 1, 1, 19, 25, 19, 1, 1, 1, 33, 55, 145, 51, 1, 1, 1, 51, 97, 595, 561, 141, 1, 1, 1, 73, 151, 1729, 2611, 2841, 393, 1, 1, 1, 99, 217, 4051, 8001, 22141, 12489, 1107, 1, 1, 1, 129, 295, 8209, 19251, 105441, 119449, 60705, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,     1,      1,      1,       1, ...
   1,   1,     1,      1,      1,       1, ...
   1,   3,     9,     19,     33,      51, ...
   1,   7,    25,     55,     97,     151, ...
   1,  19,   145,    595,   1729,    4051, ...
   1,  51,   561,   2611,   8001,   19251, ...
   1, 141,  2841,  22141, 105441,  369501, ...
   1, 393, 12489, 119449, 627873, 2319801, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000012, A002426, A084605.
Main diagonal gives A307844.
Cf. A292627, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, k^(2*j)] * Binomial[n, 2*j] * Binomial[2*j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

A(n,k) is the constant term in the expansion of (k/x + 1 + k*x)^n.
A(n,k) is the coefficient of x^n in the expansion of (1 + x + (k*x)^2)^n.
A(n,k) = Sum_{j=0..n} (1-2*k)^(n-j) * k^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (1+2*k)^(n-j) * (-k)^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k^2) * (n-1) * A(n-2,k).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 0

Offset: 0

Seiichi Manyama, May 05 2019

			Square array begins:
   1,   1,     1,     1,      1,       1,       1, ...
   0,   1,     2,     3,      4,       5,       6, ...
   0,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).

cp's OEIS Frontend

A026375 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).

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A081671 Expansion of e.g.f. exp(4x) * I_0(2x).

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A098409 Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).

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A098410 Expansion of 1/(sqrt(1-4*x)*sqrt(1-8*x)).

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A186925 Coefficient of x^n in (1+n*x+x^2)^n.

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A104454 Expansion of 1/(sqrt(1-5x)*sqrt(1-9x)).

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

A098409 Expansion of 1/(sqrt(1-3x)sqrt(1-7*x)).

A098410 Expansion of 1/(sqrt(1-4x)sqrt(1-8*x)).

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

A307847 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k^2)*x^2).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

A292628 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)BesselI(1,2*x).