A084615 -id:A084615 - cp's OEIS frontend

A006139 na(n) = 2(2n-1)a(n-1) + 4(n-1)a(n-2) with a(0) = 1.

1, 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152, 11258054144, 52767312896, 247736643584, 1164829376512, 5484233814016, 25852072517632, 121997903495168

Offset: 0

N. J. A. Sloane

a(n) = number of Delannoy paths (A001850) from (0,0) to (n,n) in which every Northeast step is immediately preceded by an East step. - David Callan, Mar 14 2004
The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
In general, 1/sqrt(1-4*r*x-4*r*x^2) has e.g.f. exp(2rx)BesselI(0,2r*sqrt((r+1)/r)x), a(n) = Sum_{k=0..n} C(2k,k)*C(k,n-k)*r^k, gives the central coefficient of (1+(2r)x+r(r+1)x^2) and is the (2r)-th binomial transform of 1/sqrt(1-8*C(n+1,2)x^2). - Paul Barry, Apr 28 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H and U steps can have two colors. - N-E. Fahssi, Feb 05 2008
Self-convolution of a(n)/2^n gives Pell numbers A000129(n+1). - Vladimir Reshetnikov, Oct 10 2016
This sequence gives the integer part of an integral approximation to Pi, and also appears in Frits Beukers's "A Rational Approach to Pi" (cf. Links, Example). Despite quality M ~ 0.9058... reported by Beukers, measurements between n = 10000 and 30000 lead to a contentious quality estimate, M ~ 0.79..., at the 99% confidence level. In "Searching for Apéry-Style Miracles" Doron Zeilberger Quotes that M = 0.79119792... and also gives a closed form. The same rational approximation to Pi also follows from time integration on a quartic Hamiltonian surface, 2*H=(q^2+p^2)*(1-4*q*(q-p)). - Bradley Klee, Jul 19 2018, updated Mar 17 2019
Diagonal of rational function 1/(1 - (x + y + x*y^2)). - Gheorghe Coserea, Aug 06 2018

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 136*x^4 + 592*x^5 + 2624*x^6 + 11776*x^7 + ...
J_3 = Integral_{y=0..Pi/4} 4*(4*(sin(y)-cos(y))*sin(y))^3*dy = 32*Pi - (304/3), |J_3| < 1. - _Bradley Klee_, Jul 19 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Frits Beukers, A rational approach to Pi, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 377.
Dario Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
Maciej Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.
Shalosh B. Ekhad and Doron Zeilberger, Searching for Apéry-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm], arXiv:1405.4445 [math.NT], 2014.
Bradley Klee, Approximating Pi with Trigonometric-Polynomial Integrals, Wolfram Demonstrations, July 27, 2018.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A001850, A002426, A036442, A084600-A084606, A084608-A084615.
Cf. A106258, A106259, A106260, A106261.
First column of A110446. A higher-quality Pi approximation: A123178.

a:=[1,2];; for n in [3..25] do a[n]:=1/(n-1)*(2*(2*n-3)*a[n-1]+4*(n-2)*a[n-2]); od; a; # Muniru A Asiru, Aug 06 2018

seq(add(binomial(2*k, k)*binomial(k, n-k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
A006139 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2):
seq(simplify(A006139(n)), n=0..29); # Peter Luschny, Sep 18 2014

Table[SeriesCoefficient[1/(1-4x-4x^2)^(1/2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 05 2012 *)
Table[Abs[LegendreP[n, I]] 2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
Table[Sum[Binomial[2*k, k]*Binomial[k, n - k], {k,0,n}], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)
a[n_] := If[n == 0, 1, Coefficient[(1 + 2 x + 2 x^2)^n, x^n]] (* Emanuele Munarini, Aug 04 2017 *)
CoefficientList[Series[1/Sqrt[(-4 x^2 - 4 x + 1)], {x, 0, 24}], x] (* Robert G. Wilson v, Jul 28 2018 *)

a(n) := coeff(expand((1+2*x+2*x^2)^n),x,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 04 2017 */

for(n=0,30,t=polcoeff((1+2*x+2*x^2)^n,n,x); print1(t","))

for(n=0,25, print1(sum(k=0,n, binomial(2*k,k)*binomial(k,n-k)), ", ")) \\ G. C. Greubel, Feb 28 2017

{a(n) = (-2*I)^n * pollegendre(n, I)}; /* Michael Somos, Aug 04 2018 */

a(n) = Sum_{k=0..n} C(2*k, k)*C(k, n-k). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
G.f.: 1/(1-4x-4x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+2x^2)^n. - Paul D. Hanna, Jun 01 2003
Inverse binomial transform of central Delannoy numbers A001850. - David Callan, Mar 14 2004
E.g.f.: exp(2*x) * BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
a(n) = Sum_{k=0..floor(n/2)} C(n,2k) * C(2k,k) * 2^(n-k). - Paul Barry, Sep 19 2006
a(n) ~ 2^(n - 3/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 05 2012, simplified Jan 31 2023
G.f.: 1/(1 - 2*x*(1+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) = 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2). - Peter Luschny, Sep 18 2014
0 = a(n)*(+16*a(n+1) + 24*a(n+2) - 8*a(n+3)) + a(n+1)*(+8*a(n+1) + 16*a(n+2) - 6*a(n+3)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Oct 13 2016
It appears that Pi/2 = Sum_{n >= 1} (-1)^(n-1)*4^n/(n*a(n-1)*a(n)). - Peter Bala, Feb 20 2017
G.f.: G(x) = (1/(2*Pi))*Integral_{y=0..2*Pi} 1/(1-x*(4*(sin(y)-cos(y))*sin(y)))*dy, also satisfies: (2+4*x)*G(x)-(1-4*x-4*x^2)*G'(x)=0. - Bradley Klee, Jul 19 2018
a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit. - Seiichi Manyama, Aug 29 2025

A084600 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x+2x^2)^n for n >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 4, 4, 1, 3, 9, 13, 18, 12, 8, 1, 4, 14, 28, 49, 56, 56, 32, 16, 1, 5, 20, 50, 105, 161, 210, 200, 160, 80, 32, 1, 6, 27, 80, 195, 366, 581, 732, 780, 640, 432, 192, 64, 1, 7, 35, 119, 329, 721, 1337, 2045, 2674, 2884, 2632, 1904, 1120, 448, 128, 1, 8, 44

Offset: 0

Paul D. Hanna, Jun 01 2003

Triangle by rows, X^n * [1,0,0,0,...]; where X = an infinite tridiagonal matrix with (1,1,1,...) in the main and subdiagonals and (2,2,2,...) in the subsubdiagonal. Also, X = an infinite triangular matrix with (1,1,2,0,0,0,...) in each column. - Gary W. Adamson, May 26 2008

Row sums = (1, 4, 16, 64, 256, ...). - Gary W. Adamson, May 26 2008

			Triangle begins:
  1;
  1, 1,  2;
  1, 2,  5,  4,   4;
  1, 3,  9, 13,  18,  12,   8;
  1, 4, 14, 28,  49,  56,  56,  32,  16;
  1, 5, 20, 50, 105, 161, 210, 200, 160,  80,  32;
  1, 6, 27, 80, 195, 366, 581, 732, 780, 640, 432, 192, 64;

Alois P. Heinz, Rows n = 0..100, flattened
G. Farkas, G. Kallos and G. Kiss, Large primes in generalized Pascal triangles, Acta Univ. Sapientiae, Informatica, 3, 2 (2011) 158-171.

Cf. A002426, A084601-A084615.

a084600 n = a084600_list !! n
a084600_list = concat $ iterate ([1,1,2] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

f:= proc(n) option remember; expand((1+x+2*x^2)^n) end:
T:= (n,k)-> coeff(f(n), x, k):
seq(seq(T(n, k), k=0..2*n), n=0..10);  # Alois P. Heinz, Apr 03 2011

t[n_, k_] := Coefficient[(1+x+2x^2)^n, x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, 2 n}] // Flatten (* Jean-François Alcover, Feb 27 2015 *)

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x^(2*k+1)*(1+x+2*x^2)/(x^(2*k+1)*(1+x+2*x^2) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013

A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

Original entry on oeis.org

1, 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265, 16786700865, 81464917185, 396215601225, 1929237099225, 9408084660945, 45928695279345, 224476389327705

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in four colors. - N-E. Fahssi, Mar 30 2008
Ignoring initial term, equals the logarithmic derivative of A091147. - Paul D. Hanna, Dec 08 2018
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. - Peter Bala, Jan 10 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1433 (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.
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.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Cf. A002426, A084600-A084604, A084606-A084615.

Cf. A322240 (a(n)^2), A091147.

a := n -> simplify(2^n*GegenbauerC(n,-n, -1/4)):
seq(a(n), n=0..22); # Peter Luschny, May 08 2016

Table[n!*SeriesCoefficient[E^x*BesselI[0,4*x],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 16];
Table[a[n], {n, 0, 22}] (* Peter Luschny, Mar 18 2018 *)

for(n=0,30,t=polcoeff((1+x+4*x^2)^n,n,x); print1(t","))
for(n=0,20,print1(a(n),", "))

{a(n) = sum(k=0,n, (-3)^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Dec 09 2018

E.g.f.: exp(x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003
a(n) is also the central coefficient of (4+x+x^2)^n; a(n) = Sum_{k=0..n} 3^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907. - N-E. Fahssi, Mar 30 2008
a(n) = (1/Pi)*integral(x=-2..2, (2*x+1)^n/sqrt((2-x)*(2+x))). - Peter Luschny, Sep 12 2011
D-finite with recurrence a(n+2) = ((2*n+3)*a(n+1) + 15*(n+1)*a(n))/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012
a(n) ~ 5^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 2^n*GegenbauerC(n, -n, -1/4). - Peter Luschny, May 08 2016
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 16). - Peter Luschny, Mar 18 2018
a(n) = Sum_{k=0..n} (-3)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k). - Paul D. Hanna, Dec 09 2018
a(n) = Sum_{k=0..n} 5^(n-k) * (-2)^k * binomial(n,k)*binomial(2*k,k). - Seiichi Manyama, May 01 2019
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A084601 Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

Original entry on oeis.org

1, 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693, 243954113, 906758785, 3375949829, 12587460557, 46995614449, 175669746209, 657370655045, 2462383495357, 9232029156001

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U (or D) can have 2 colors. - N-E. Fahssi, Feb 05 2008
Diagonal of the rational function 1 / (1 - 2*x^2 - y^2 - x*y). - Ilya Gutkovskiy, Apr 22 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1718 (terms 0..120 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.

Cf. A002426, A084600, A084602-A084615.

Cf. A007318, A027907, A036442.

a := n -> hypergeom([1/2 - n/2, -n/2], [1], 8):
seq(simplify(a(n)), n=0..24); # Peter Luschny, Mar 18 2018

CoefficientList[Series[1/Sqrt[1-2x-7x^2],{x,0,30}],x] (* Harvey P. Dale, Sep 18 2011 *)

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

for(n=0,30,t=polcoeff((1+x+2*x^2)^n,n,x); print1(t","))

E.g.f.: exp(x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)*2^k. - Paul Barry, Aug 26 2004
a(n) = Sum_{k=0..n} Trinomial(k, n)*Binomial(n, k), with Trinomial=A027907 and Binomial=A007318. - Ralf Stephan, Jan 28 2005
a(n) is also the central coefficient of (2+x+x^2)^n; a(n) = Sum_{k=0..n} C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = coefficient of x^n of (1+x+x^2)^k : A027907 - N-E. Fahssi, Feb 05 2008
a(n+2) = ((2*n+3)*a(n+1) + 7*(n+1)*a(n))/(n+2); a(0)=a(1)=1. - Sergei N. Gladkovskii, Aug 01 2012
G.f.: G(0), where G(k)= 1 + x*(2+7*x)*(4*k+1)/( 4*k+2 - x*(2+7*x)*(4*k+2)*(4*k+3)/(x*(2+7*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
a(n) ~ sqrt(8+2*sqrt(2)) * (1+2*sqrt(2))^n / (4*sqrt(Pi*n)). - Vaclav Kotesovec, May 09 2014
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 8). - Peter Luschny, Mar 18 2018

A084609 Coefficients of 1/sqrt(1-4x-8x^2); also, a(n) is the central coefficient of (1+2x+3x^2)^n.

Original entry on oeis.org

1, 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424, 78210099718, 414862155980, 2204273582236, 11729283976136, 62496686731924, 333400654676168

Offset: 0

Paul D. Hanna, Jun 01 2003

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U can have 3 colors and H can have 2 colors. - N-E. Fahssi, Mar 30 2008
Self-convolution of a(n)/2^n gives A002605(n+1). - Vladimir Reshetnikov, Oct 10 2016
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. - Peter Bala, Jan 07 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1358 (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.

Cf. A002426, A084600 - A084608, A084610 - A084615.

Row sums of A328347.

A084609:= func< n | (&+[Binomial(n,j)*Binomial(2*(n-j),n)*2^j: j in [0..Floor(n/2)]]) >;
[A084609(n): n in [0..50]]; // G. C. Greubel, Mar 26 2023

(* Programs from Robert G. Wilson v, Mar 02 2011 *)
a[n_]:= Sum[Binomial[n, k] Binomial[2(n-k), n] 2^k, {k, 0, n/2}]; Array[a, 30, 0]
a[n_]:= CoefficientList[Expand[(1 +2x +3x^2)^n], x][[n+1]]; Array[a, 30, 0]
CoefficientList[Series[1/Sqrt[1 -4x -8x^2], {x,0,30}], x]
Range[0, 30]! CoefficientList[ Series[ Exp[ 2x] BesselI[0, Sqrt[12] x], {x, 0, 30}], x] (* End *)
Table[2^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 3], {n,0,30}] (* Vladimir Reshetnikov, Oct 10 2016 *)

a(n):=coeff(expand((1+2*x+3*x^2)^n),x,n);
makelist(a(n),n,0,12);

for(n=0,30,t=polcoeff((1+2*x+3*x^2)^n,n,x); print1(t","))

def A084609(n): return sum(binomial(n,j)*binomial(2*(n-j),n)*2^j for j in range(n//2+1))
[A084609(n) for n in range(51)] # G. C. Greubel, Mar 26 2023

a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(2*(n-k),n)*2^k. - Paul Barry, Sep 08 2004
a(n) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*3^k*2^(n-2*k); a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(n-k,k)*3^k*2^(n-2k). - Paul Barry, Sep 19 2006
E.g.f.: exp(2*x) * Bessel_I(0,2*sqrt(3)*x)
a(n) = ( 2*(2*n-1)*a(n-1) + 8*(n-1)*a(n-2) )/n, a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 20 2012
a(n) ~ sqrt(18+6*sqrt(3))*(2+2*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - 2*x*(1+2*x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+2*x)/(k+1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: G(0), where G(k)= 1 + x*(2+4*x)*(4*k+1)/(2*k+1 - x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(1+2*x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 18 2013
a(n) = 2^n * hypergeom([(1-n)/2,-n/2], [1], 3) = binomial(2*n, n) * hypergeom([(1-n)/2,-n/2], [1/2-n], -2). - Vladimir Reshetnikov, Oct 10 2016
a(n) = (-2*sqrt(-2))^n * P(n, sqrt(-1/2)), where P(n,x) denotes the n-th Legendre polynomial. - Peter Bala, Feb 07 2022

A084610 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -2, 1, 1, 3, 0, -5, 0, 3, -1, 1, 4, 2, -8, -5, 8, 2, -4, 1, 1, 5, 5, -10, -15, 11, 15, -10, -5, 5, -1, 1, 6, 9, -10, -30, 6, 41, -6, -30, 10, 9, -6, 1, 1, 7, 14, -7, -49, -14, 77, 29, -77, -14, 49, -7, -14, 7, -1, 1, 8, 20, 0, -70, -56, 112, 120, -125, -120, 112, 56, -70, 0, 20, -8, 1

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
  1;
  1, 1, -1;
  1, 2, -1,  -2,   1;
  1, 3,  0,  -5,   0,   3, -1;
  1, 4,  2,  -8,  -5,   8,  2,  -4,   1;
  1, 5,  5, -10, -15,  11, 15, -10,  -5,   5, -1;
  1, 6,  9, -10, -30,   6, 41,  -6, -30,  10,  9, -6,   1;
  1, 7, 14,  -7, -49, -14, 77,  29, -77, -14, 49, -7, -14, 7, -1;

Paul D. Hanna, Rows n=0..34 of triangle, flattened
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 13.

Cf. A002426, A057597, A084600 - A084609, A084611 - A084615.

A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;
[A084610(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 26 2023

T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-1)^j, {j,0,k}];
Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 26 2023 *)

for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-x^2)^n,k,x); print1(t",")); print(" "))

def A084610(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-1)^j for j in range(k+1))
flatten([[A084610(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 26 2023

G.f.: G(0)/2 , where G(k)= 1 + 1/( 1 - (1+x-x^2)*x^(2*k+1)/((1+x-x^2)*x^(2*k+1) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
From G. C. Greubel, Mar 26 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-1)^j.
T(n, 2*n) = (-1)^n.
T(n, 2*n-1) = (-1)^(n-1)*n, n >= 1.
Sum_{k=0..2*n} T(n, k) = 1.
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-1)^n.
Sum_{k=0..n} T(n-k, k) = floor((n+2)/2).
Sum_{k=0..n} (-1)^k*T(n-k, k) = (-1)^n*A057597(n+2). (End)

A084603 Coefficients of 1/sqrt(1 - 2x - 11x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

Original entry on oeis.org

1, 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139, 2778852979, 12043194163, 52286516821, 227323871929, 989675651041, 4313712072241, 18822940658947, 82215245701519

Offset: 0

Paul D. Hanna, Jun 01 2003

5th binomial transform of 2^n*LegendreP(n,-2) (signed version of A069835). - Paul Barry, Sep 03 2004
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in three colors. - N-E. Fahssi, Feb 05 2008
Diagonal of the rational function 1 / (1 - 3*x^2 - y^2 - x*y). - Ilya Gutkovskiy, Apr 22 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1542 (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.

Cf. A002426, A084600-A084602, A084604-A084615, A025237.

Table[Sum[Binomial[n-k,k]*Binomial[n,k]*3^k,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

for(n=0,30,t=polcoeff((1+x+3*x^2)^n,n,x); print1(t","))

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)3^k. - Paul Barry, Aug 26 2004
Binomial transform is A084609. Hankel transform is 6^n*3^C(n,2). - Paul Barry, Sep 16 2006
a(n) = (1/Pi)*Integral_{x=1-2*sqrt(3)..1+2*sqrt(3)} x^n/sqrt(-x^2 + 2*x + 11). - Paul Barry, Sep 16 2006
From Paul Barry, Sep 16 2006: (Start)
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*3^k;
a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k,k)*3^k. (End)
From N-E. Fahssi, Feb 05 2008: (Start)
a(n) is also the central coefficient of (3+x+x^2)^n;
a(n) = Sum_{k=0..n} 2^(n-k)*C(n,k)*T(k,n), where T(k,n) is the triangle of trinomial coefficients = coefficient of x^n of (1+x+x^2)^k: A027907. (End)
D-finite with recurrence: a(n+2) = ( (2*n+3)*a(n+1) + 11*(n+1)*a(n) )/(n+2); a(0)=a(1)=1. - Sergei N. Gladkovskii, Aug 01 2012
a(n) ~ sqrt(18+3*sqrt(3))*(1+2*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
E.g.f.: exp(x)*BesselI(0, 2*sqrt(3)*x). - Paul D. Hanna, Nov 09 2014, after Vladeta Jovovic in A084601
From Peter Bala, Jan 07 2022: (Start)
O.g.f. A(x) = 1 + x*d/dx(log(B(x))), where B(x) = (1 - x - sqrt(1 - 2*x - 11*x^2))/(6*x^2) is the o.g.f. of A025237.
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)

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 10, 12, 9, 1, 6, 21, 44, 63, 54, 27, 1, 8, 36, 104, 214, 312, 324, 216, 81, 1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243, 1, 12, 78, 340, 1095, 2712, 5284, 8136, 9855, 9180, 6318, 2916, 729, 1, 14, 105, 532, 2009, 5922, 13993, 26840, 41979

Offset: 0

Paul D. Hanna, Jun 01 2003

			Triangle begins:
  1;
  1,  2,  3;
  1,  4, 10,  12,   9;
  1,  6, 21,  44,  63,   54,   27;
  1,  8, 36, 104, 214,  312,  324,  216,   81;
  1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243;

Alois P. Heinz, Rows n = 0..100, flattened

Cf. A000079, A000244, A000400, A002426, A084600 - A084606, A006139, A084609 - A084615, A101822, A212697.

a084608 n = a084608_list !! n
a084608_list = concat $ iterate ([1,2,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

A084608:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*2^(k-2*j)*3^j: j in [0..k]]) >;
[A084608(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 27 2023

f:= proc(n) option remember; expand((1+2*x+3*x^2)^n) end:
T:= (n,k)-> coeff(f(n), x, k):
seq(seq(T(n, k), k=0..2*n), n=0..10);  # Alois P. Heinz, Apr 03 2011

row[n_] := (1+2x+3x^2)^n + O[x]^(2n+1) // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+2*x+3*x^2)^n,k,x); print1(t",")); print(" "))

def A084608(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*2^(k-2*j)*3^j for j in range(k//2+1))
flatten([[A084608(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 27 2023

From G. C. Greubel, Mar 27 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*2^(k-2*j)*3^j.
T(n, n) = A084609(n).
T(n, 2*n-1) = A212697(n), n >= 1.
T(n, 2*n) = A000244(n).
Sum_{j=0..2*n} T(n, k) = A000400(n).
Sum_{k=0..2*n} (-1)^k*T(n, k) = A000079(n).
Sum_{k=0..n} T(n-k, k) = A101822(n). (End)

A084611 a(n) = sum of absolute values of coefficients of (1+x-x^2)^n.

Original entry on oeis.org

1, 3, 7, 13, 35, 83, 165, 367, 899, 1957, 3839, 9771, 22709, 43213, 102963, 255061, 525601, 1098339, 2798273, 6202969, 11746259, 29976073, 70898649, 140495779, 314391789, 787757461, 1688887719, 3337986541, 8583687613, 19647782463

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

Limit_{n -> oo} a(n+1)/a(n) does not exist; however, lim_{n -> oo} a(n)^(1/n) = sqrt(5) (conjecture).

Paul D. Hanna, Table of n, a(n) for n = 0..500
Vaclav Kotesovec, Asymptotic of sequence A084611, Jul 26 2013.

Cf. A002426, A084600 - A084610, A084612 - A084615.

A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;
[(&+[Abs(A084610(n,k)): k in [0..2*n]]): n in [0..50]]; // G. C. Greubel, Mar 26 2023

Table[Sum[Abs[Coefficient[Expand[(1+x-x^2)^n],x,k]],{k,0,2*n}],{n,0,30}] (* Vaclav Kotesovec, Jul 28 2013 *)

{a(n)=sum(k=0,2*n,abs(polcoeff((1+x-x^2+x*O(x^k))^n,k)))}
for(n=0,30,print1(a(n),", "))

def A084610(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*(-1)^j for j in range(k//2+1))
def A084611(n): return 2*sum(abs(A084610(n,k)) for k in range(n)) + abs(A084610(n,n))
[A084611(n) for n in range(50)] # G. C. Greubel, Mar 26 2023

cp's OEIS Frontend

A006139 n*a(n) = 2*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) with a(0) = 1.

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A084600 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x+2x^2)^n for n >= 0.

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A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

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A084601 Coefficients of 1/(1-2*x-7*x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

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A084609 Coefficients of 1/sqrt(1-4*x-8*x^2); also, a(n) is the central coefficient of (1+2*x+3*x^2)^n.

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A084610 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.

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A006139 na(n) = 2(2n-1)a(n-1) + 4(n-1)a(n-2) with a(0) = 1.

A084601 Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

A084609 Coefficients of 1/sqrt(1-4x-8x^2); also, a(n) is the central coefficient of (1+2x+3x^2)^n.

A084603 Coefficients of 1/sqrt(1 - 2x - 11x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.