A084610 -id:A084610 - cp's OEIS frontend

A104509 Matrix inverse of triangle A104505, which is the right-hand side of triangle A084610 of coefficients in (1 + x - x^2)^n.

1, 1, -1, 3, -2, 1, 4, -6, 3, -1, 7, -12, 10, -4, 1, 11, -25, 25, -15, 5, -1, 18, -48, 60, -44, 21, -6, 1, 29, -91, 133, -119, 70, -28, 7, -1, 47, -168, 284, -296, 210, -104, 36, -8, 1, 76, -306, 585, -699, 576, -342, 147, -45, 9, -1, 123, -550, 1175, -1580, 1485, -1022, 525, -200, 55, -10, 1, 199, -979, 2310, -3454, 3641

Offset: 0

Paul D. Hanna, Mar 11 2005

Riordan array ( (1 + x^2)/(1 - x - x^2), -x/(1 - x - x^2) ) belonging to the hitting time subgroup of the Riordan group (see Peart and Woan). - Peter Bala, Jun 29 2015

The sums of absolute values along steep diagonals in this triangle are: 1, 1, 3, 4 + |-1|, 7 + |-2|, 11 + |-6|, 18 + |-12| + 1, ... and these are the tribonacci numbers A000213 that begin with 1, 1, 1, 3. To see this, replace the y in the g.f. A(x,y) = (1 + x^2)/(1-x-x^2 + x*y) with y=-x^2, multiply by x, and add 1, to obtain the g.f. (1 - x^2)/(1-x-x^2-x^3) for A000213. - Noah Carey and Greg Dresden, Nov 02 2021

			Rows begin:
   1;
   1,   -1;
   3,   -2,   1;
   4,   -6,   3,   -1;
   7,  -12,  10,   -4,   1;
  11,  -25,  25,  -15,   5,   -1;
  18,  -48,  60,  -44,  21,   -6,   1;
  29,  -91, 133, -119,  70,  -28,   7,  -1;
  47, -168, 284, -296, 210, -104,  36,  -8, 1;
  76, -306, 585, -699, 576, -342, 147, -45, 9, -1; ...

Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0 to 141, flattened).
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Wikipedia, Lucas polynomials.

Leftmost column is A000204 (Lucas numbers). Other columns include: A045925, A067988. Row sums are: {1,0,2,0,2,0,2,...}. Absolute row sums form: A099425. Antidiagonal sums are: {1,1,2,2,2,2,2,...}. Absolute antidiagonal sums are: A084214.

Cf. A104505, A084610, A000213.

S:= series((1 + x^2)/(1-x-x^2 + x*y),x, 20):
for n from 0 to 19 do R[n]:= coeff(S,x,n) od:
seq(seq(coeff(R[n],y,j),j=0..n), n=0..19); # Robert Israel, Jun 30 2015

nmax = 11;
T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
M = Table[T[n, k], {n, 0, nmax}, {k, 0, nmax}] // Inverse;
Table[M[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)

{ T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1 + X^2)/(1-X-X^2 + X*Y),n,x),k,y); }

{ tabl(nn) = my(m = matrix(nn, nn, n, k, n--; k--; if((nMichel Marcus, Jun 30 2015

{ A104509(n,k) = if(n==0, k==0, (-1)^k * sum(i=0, (n-k)\2, n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k) )); } \\ Max Alekseyev, Oct 11 2021

For n>=1, a(n,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-i,i) * binomial(n-2*i,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k). - Max Alekseyev, Oct 11 2021
G.f.: A(x, y) = (1 + x^2)/(1-x-x^2 + x*y).
G.f. for column k: g_k(x) = -(x^2+1)*x^k/(x^2+x-1)^(k+1). - Robert Israel, Jun 30 2015
G.f. for row n>=1 is the Lucas polynomial L_n(1-x). - Max Alekseyev, Oct 11 2021

A104505 Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.

Original entry on oeis.org

1, 1, -1, -1, -2, 1, -5, 0, 3, -1, -5, 8, 2, -4, 1, 11, 15, -10, -5, 5, -1, 41, -6, -30, 10, 9, -6, 1, 29, -77, -14, 49, -7, -14, 7, -1, -125, -120, 112, 56, -70, 0, 20, -8, 1, -365, 117, 288, -126, -126, 90, 12, -27, 9, -1, -131, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1, 1409, 946, -1265, -495, 858, 33, -363, 110, 55, -44

Offset: 0

Paul D. Hanna, Mar 11 2005

Matrix inverse is triangle A104509 and is related to Fibonacci numbers. Column 0 equals A098331, with g.f.: 1/sqrt(1-2*x+5*x^2). Column 1 equals A104506, with g.f.: ((1-x)/sqrt(1-2*x+5*x^2)-1)/(2*x). Row sums equal A104507. Absolute row sums equal A104508.

Array (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x)), in Riordan array notation. Product of A120616 by A007318. - Paul Barry, Jun 17 2006

			Rows begin:
1;
1,-1;
-1,-2,1;
-5,0,3,-1;
-5,8,2,-4,1;
11,15,-10,-5,5,-1;
41,-6,-30,10,9,-6,1;
29,-77,-14,49,-7,-14,7,-1;
-125,-120,112,56,-70,0,20,-8,1;
-365,117,288,-126,-126,90,12,-27,9,-1;
-131,770,45,-540,90,228,-105,-30,35,-10,1; ...

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A104509, A098331, A007440, A104506, A104507, A104508, A108044.

T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)

PARI
```
T(n,k)=if(n
				
```

T(n, 0) = A098331(n). T(n, 1) = n*A007440(n) (n>0).
Column k has e.g.f. exp(x)*Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry, Jun 17 2006
From Peter Bala, Jun 29 2015: (Start)
Matrix factorization in the Riordan group: ( 1/(1 - x), x/(1 - x) ) * ( 1/sqrt(1 + 4*x^2), (1 - sqrt(1 + 4*x^2))/(2*x) ) = A007318 * signed version of A108044.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x - sqrt(1 - 2*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = x^2 + x - 1. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A104507 Row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

Original entry on oeis.org

1, 0, -2, -3, 2, 15, 19, -28, -134, -129, 353, 1254, 791, -4238, -11818, -3123, 49162, 110007, -17783, -554458, -996323, 690932, 6096792, 8624747, -12287153, -65419110, -69285296, 178655307, 684550946, 483569751, -2354830741, -6970706252, -2324044054, 29195280375, 68793790705

Offset: 0

Paul D. Hanna, Mar 11 2005

Cf. A104505, A1086223, A246437, A370616, A386548.

CoefficientList[Series[(x/((1 - x)) + 1/((-Sqrt[5 x^2 - 2 x + 1] + x + 1)) x (1 - (5 x - 1)/(Sqrt[5 x^2 - 2 x + 1]))), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 05 2015 *)

a(n):=sum((-1)^j*binomial(n,j)*binomial(n-j-1,n-2*j),j,0,n/2); /* Vladimir Kruchinin, Oct 04 2015 */

a(n)=sum(k=0,n,polcoeff((1+x-x^2)^n,n+k))

a(n) = sum(k=0, n/2, (-1)^k*binomial(n,k)*binomial(n-k-1,n-2*k));
vector(40, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

G.f.: (x/((1-x))+1/((-sqrt(5*x^2-2*x+1)+x+1))*x*(1-(5*x-1)/(sqrt(5*x^2-2*x+1)))). - Vladimir Kruchinin, Oct 04 2015
a(n) = Sum_{j=0..n/2}((-1)^j*binomial(n,j)*binomial(n-j-1,n-2*j)). - Vladimir Kruchinin, Oct 04 2015
From Peter Bala, Jul 24 2025: (Start)
a(n) = [x^n] (1 - x^2/(1 - x))^n. Cf. A246437.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all primes p and all positive integers n and k.
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 - x^2 - x^3 + x^4 + 4*x^5 + ... is the g.f. of A108623.(End)

A104508 Absolute row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

Original entry on oeis.org

1, 2, 4, 9, 20, 47, 103, 198, 512, 1161, 1985, 5590, 13005, 21684, 59294, 142273, 270858, 634187, 1526399, 3256372, 6779657, 16020364, 37758202, 79702477, 164647757, 424655100, 940639154, 1709638551, 4675803604, 10774789429

Offset: 0

Paul D. Hanna, Mar 11 2005

nonn

Cf. A104505.

a(n)=sum(k=0,n,abs(polcoeff((1+x-x^2)^n,n+k)))

A104506 Column 1 of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1 + x - x^2)^n.

Original entry on oeis.org

0, -1, -2, 0, 8, 15, -6, -77, -120, 117, 770, 946, -1728, -7735, -6930, 22800, 76960, 42245, -282150, -751640, -125800, 3341205, 7145710, -2002725, -38228232, -65418925, 55550014, 424605078, 566938400, -936604097, -4587287310

Offset: 0

Paul D. Hanna, Mar 11 2005

sign

P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, page 8.

Cf. A104505, A007440, A104507, A104508, A104509.

{a(n)=polcoeff(((1-x)/sqrt(1-2*x+5*x^2+x^2*O(x^n))-1)/(2*x),n)}

G.f.: ((1-x)/sqrt(1-2*x+5*x^2) - 1)/(2*x).
a(n) = (-1)^n*n*A007440(n) (reversion of g.f. for Fibonacci numbers).
a(n) = -Sum_{k=0..floor(n/2)} C(n, k)*C(n-k, k+1)*(-1)^k. - Paul Barry, May 02 2005
E.g.f.: -exp(x)Bessel_I(1,2*i*x)/i, i=sqrt(-1). - Paul Barry, Feb 10 2006
-(n-1)*(n+1)*a(n) + n*(2*n-1)*a(n-1) - 5*n*(n-1)*a(n-2) = 0. - R. J. Mathar, Aug 17 2017

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

Original entry on oeis.org

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

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

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

A078996 Triangle read by rows: let f(x) = x/(1-x-x^2); n-th row gives coefficients of denominator polynomial of n-th derivative f(x)^(n), with highest powers first, for n >= 0.

Original entry on oeis.org

-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

Mohammad K. Azarian, Jan 12 2003

			Triangle begins:
  -1, -1,  1;
   1,  2, -1, -2,  1;
   1,  3,  0, -5,  0,  3, -1;
  ...

See A084610 for another version of this triangle.

f(x)^(n), for n=0, 1, 2, 3, 4, ..., where f(x)= x/(1-x-x^2).

G.f.: G(0)/(2*x) - 1/x - 2 - 2*x + 2*x^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

Edited by N. J. A. Sloane, Jan 15 2011

A137454 Irregular triangle read by rows: coefficients of polynomials p(x, n) where p(x,n) = x^2p(x,n-1) + (-x-1)p(x,n-2) + p(x,n-3).

Original entry on oeis.org

1, -1, -1, 1, -1, -1, -1, -1, 1, 2, 2, -1, -2, -1, -1, 1, 0, 1, 5, 4, -1, -3, -1, -1, 1, -3, -5, -2, 3, 9, 6, -1, -4, -1, -1, 1, 2, 1, -10, -16, -6, 6, 14, 8, -1, -5, -1, -1, 1, 3, 9, 14, 4, -23, -34, -12, 10, 20, 10, -1, -6, -1, -1, 1, -5, -8, 10, 38, 45, 10, -44, -60, -20, 15, 27, 12, -1, -7, -1, -1, 1, -1, -11, -38, -42, 23, 101, 105, 20

Offset: 0

Roger L. Bagula and Gary W. Adamson, Apr 18 2008

			{1},
{-1, -1, 1},
{-1, -1, -1, -1, 1},
{2, 2, -1, -2, -1, -1, 1},
{0, 1, 5, 4, -1, -3, -1, -1, 1},
{-3, -5, -2, 3,9, 6, -1, -4, -1, -1, 1},
{2, 1, -10, -16, -6, 6,14, 8, -1, -5, -1, -1, 1},
{3, 9,14, 4, -23, -34, -12, 10, 20, 10, -1, -6, -1, -1, 1},
{-5, -8, 10, 38, 45, 10, -44, -60, -20, 15,27, 12, -1, -7, -1, -1, 1},
{-1, -11, -38, -42, 23, 101, 105,20, -75, -95, -30, 21, 35, 14, -1, -8, -1, -1,1},
{8, 22, 11, -55, -144, -131, 45, 215, 205, 35, -118, -140, -42, 28, 44, 16, -1, -9, -1, -1, 1}

Cf. A084610.

p[x, -1] = 0;
p[x, 0] = 1;
p[x, 1] = x^2 - x - 1;
p[x_, n_] := x^2*p[x, n - 1] + (-x - 1)*p[x, n - 2] + p[x, n - 3];
Table[Expand[p[x, n]], {n, 0, 10}];
a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[a1]

p(x,-1)=0; p(x,0)=1; p(x,1) = x^2 - x - 1; p(x,n) = x^2*p(x,n-1) + (-x-1)*p(x,n-2) + p(x,n-3).

Heavily edited and corrected by Joerg Arndt, Apr 30 2018

cp's OEIS Frontend

A104509 Matrix inverse of triangle A104505, which is the right-hand side of triangle A084610 of coefficients in (1 + x - x^2)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A104505 Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A104507 Row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Maxima

PARI

PARI

Formula

A104508 Absolute row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

Views

Author

Keywords

Crossrefs

Programs

PARI

A104506 Column 1 of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1 + x - x^2)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

Maxima

PARI

PARI

PARI

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

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

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

A137454 Irregular triangle read by rows: coefficients of polynomials p(x, n) where p(x,n) = x^2p(x,n-1) + (-x-1)p(x,n-2) + p(x,n-3).