A084601 -id:A084601 - cp's OEIS frontend

A036442 a(n) = 2^((n-1)*(n+2)/2).

1, 4, 32, 512, 16384, 1048576, 134217728, 34359738368, 17592186044416, 18014398509481984, 36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, 20282409603651670423947251286016, 664613997892457936451903530140172288

Offset: 1

Abdallah Rayhan (rayhan(AT)engr.uvic.ca)

Number of redundant paths for a fault-tolerant ATM switch.
Hankel transform (see A001906 for definition ) of A001850, A006139, A084601; also Hankel transform of the sequence 1, 0, 4, 0, 24, 0, 160, 0, 1120, ... (A059304 with interpolated zeros). - Philippe Deléham, Jul 03 2005
Hankel transform of A109980. Unsigned version of A127945. - Philippe Deléham, Dec 11 2008
a(n) = the multiplicative Wiener index of the wheel graph with n+3 vertices. The multiplicative Wiener index of a connected simple graph G is defined as the product of the distances between all pairs of distinct vertices of G. The wheel graph with n+3 vertices has (n+3)(n+2)/2 pairs of distinct vertices, of which 2(n+2) are adjacent; each of the remaining (n+2)(n-1)/2 pairs are at distance 2; consequently, the multiplicative Wiener index is 2^((n-1)(n+2)/2) = a(n). - Emeric Deutsch, Aug 17 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..80
P. Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161, 2013 and J. Int. Seq. 17 (2014) # 14.5.1
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
C. Lo and C. Chiu, A Fault-Tolerant Architecture for ATM Networks, 20th IEEE Conf. Local Computer Networks, 1995, pp. 29-36
Index to divisibility sequences

I:=[1]; [n le 1 select I[n] else Self(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Oct 24 2012

Table[2^((n-1) * (n+2)/2), {n, 1, 30}] (* Vincenzo Librandi, Oct 24 2012 *)

A036442[n]:=2^((n-1)*(n+2)/2)$
makelist(A036442[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

a(n)=2^((n-1)*(n+2)/2) \\ Charles R Greathouse IV, Oct 24 2012

a(1) = 1, a(n) = a(n-1) * 2^n. - Vincenzo Librandi, Oct 24 2012

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

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)

A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

Original entry on oeis.org

1, 1, 11, 31, 211, 851, 4901, 22961, 124531, 623011, 3313201, 17086301, 90453661, 473616781, 2509264811, 13250049551, 70368250451, 373539254611, 1989045489281, 10597110956861, 56566637447401, 302196871378601, 1616570627763311, 8654955238504531, 46384344189261661

Offset: 0

Paul Barry, Aug 31 2004

Central coefficient of (1+x+5x^2)^n.

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have five colors. - N-E. Fahssi, Mar 31 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A084601, A084603, A084605.

Table[SeriesCoefficient[1/Sqrt[1-2*x-19*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

x='x+O('x^66); Vec(1/(1-2*x-19*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

E.g.f. : exp(x)*BesselI(0, 2*sqrt(5)x).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*5^k}.
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*5^k}.
n*a(n) +(1-2*n)*a(n-1) +19*(1-n)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(50+5*sqrt(5))*(1+2*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

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).

A014431 a(1) = 1, a(2) = 2, a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

Original entry on oeis.org

1, 2, 2, 6, 14, 42, 122, 382, 1206, 3922, 12914, 43190, 145950, 498170, 1714026, 5940014, 20712646, 72623266, 255875298, 905477734, 3216853294, 11469069258, 41023019098, 147166210014, 529374272470, 1908965352434, 6899707805522, 24991194656022, 90698707816766

Offset: 1

Clark Kimberling and Wouter Meeussen

Michael De Vlieger, Table of n, a(n) for n = 1..1724
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A025235, A006318, A084601.

a:=[1,2]; for n in [3..30] do Append(~a,&+[a[k]*a[n-k]:k in [1..n-2]] ); end for; a; // Marius A. Burtea, Jan 02 2020

Rest@ CoefficientList[Series[(1 + x - Sqrt[1 - 2 x - 7 x^2])/2, {x, 0, 27}], x] (* Michael De Vlieger, Jan 02 2020 *)

a(n)=polcoeff((1+x-sqrt(1-2*x-7*x^2+x*O(x^n)))/2,n)

a(n) = 2*A025235(n-2) for n>=2.
G.f.: (1+x-sqrt(1-2*x-7*x^2))/2. - Michael Somos, Jun 08 2000
a(n) = (A084601(n) - A084601(n-1))/(2*(n-1)) for n > 1. - Mark van Hoeij, Jul 02 2010
G.f.: x + 2*x^2/G(0) with G(k) = (1 - x - 2*x^2/G(k+1)) (continued fraction). - Nikolaos Pantelidis, Dec 16 2022
From Peter Bala, May 01 2024: (Start)
O.g.f.: A(x) = x*S(x/(1 + 2*x)) = 2*x - x*S(- x/(1 - 4*x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.
The g.f. satisfies A(x)^2 - (1 + x)*A(x) + x*(1 + 2*x) = 0.
A(x) = x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - ...))).
A(x) = x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - ...))))))). (End)
D-finite with recurrence n*a(n) +(-2*n+3)*a(n-1) +7*(-n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

Corrected by T. D. Noe, Oct 31 2006

A110180 Triangle of generalized central trinomial coefficients.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jul 14 2005

Rows sums are A110181. Diagonal sums are A110182. Columns include central trinomial coefficients A002426, A084601, A084603, A084605, A098264. T(n,k) = central coefficient (1 + x + kx^2)^n.

			Rows begin
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  7,  5,  1,  1;
  1, 19, 13,  7,  1,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n - k - j, j]*k^j, {j, 0, Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Mar 05 2017 *)

Number triangle T(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(n-k-j, j)*k^j.

A084602 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 3x^2)^n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 7, 6, 9, 1, 3, 12, 19, 36, 27, 27, 1, 4, 18, 40, 91, 120, 162, 108, 81, 1, 5, 25, 70, 185, 331, 555, 630, 675, 405, 243, 1, 6, 33, 110, 330, 726, 1441, 2178, 2970, 2970, 2673, 1458, 729, 1, 7, 42, 161, 539, 1386, 3157, 5797, 9471, 12474, 14553

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,1, 3},
{1,2, 7,  6,  9},
{1,3,12, 19, 36, 27,  27},
{1,4,18, 40, 91,120, 162, 108,  81},
{1,5,25, 70,185,331, 555, 630, 675, 405, 243},
{1,6,33,110,330,726,1441,2178,2970,2970,2673,1458,729},

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002426, A084600-A084601, A084603-A084615.

With[{eq = (1 + x + 3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n, 0, 10}]]] (* G. C. Greubel, Mar 02 2017 *)

for(n=0,15, for(k=0,2*n,t=polcoeff((1+x+3*x^2)^n,k,x); print1(t",")); print(" "))

A098265 G.f. : 1/(1-2x-23x^2)^(1/2).

Original entry on oeis.org

1, 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357, 232260610177, 1327920945601, 7627285093069, 43787832627493, 252042452907169, 1451244932278129, 8370001674641917, 48303478743113893, 279083099450496961

Offset: 0

Paul Barry, Aug 31 2004

Central coefficient of (1+x+6x^2)^n.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A084601, A084603, A084605, A098264.

Table[SeriesCoefficient[1/Sqrt[1-2*x-23*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

x='x+O('x^66); Vec(1/(1-2*x-23*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(x)*BesselI(0, 2*sqrt(6)x).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*6^k}.
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*6^k}.
n*a(n) +(1-2n)*a(n-1) +23(1-n)*a(n-2)=0. (Recurrence (4) in the Noe paper).- Veka Gesell, Jun 26 2012
a(n) ~ sqrt(72+6*sqrt(6))*(1+2*sqrt(6))^n/(12*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)C(j,n-j)2^(n-j).

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 13, 9, 5, 1, 49, 31, 17, 7, 1, 161, 105, 61, 29, 9, 1, 581, 371, 217, 111, 45, 11, 1, 2045, 1313, 781, 417, 189, 65, 13, 1, 7393, 4719, 2825, 1551, 753, 303, 89, 15, 1, 26689, 17041, 10277, 5757, 2921, 1289, 461, 117, 17, 1

Offset: 0

Paul Barry, Feb 10 2006

First column is A084601 with e.g.f. exp(x) Bessel_I(0,2*sqrt(2)x). Row sums are A098518(n+1) with e.g.f. dif(exp(x) Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

Riordan array (1/sqrt(1-2*x-7*x^2), (1+x-sqrt(1-2*x-7*x^2))/2).

			Triangle begins as:
    1;
    1,   1;
    5,   3,   1;
   13,   9,   5,   1;
   49,  31,  17,   7,  1;
  161, 105,  61,  29,  9,  1;
  581, 371, 217, 111, 45, 11, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A084601 (k=0), A098518.

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j)) ))); # G. C. Greubel, May 09 2019

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

A115991 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j)*2^(n-j),j=0..n) ;
end proc:
seq(seq(A115991(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Table[Sum[Binomial[n-k, j-k]*Binomial[j, n-j]*2^(n-j), {j, 0, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 09 2019 *)

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j))}; \\ G. C. Greubel, May 09 2019

[[sum(binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

cp's OEIS Frontend

A036442 a(n) = 2^((n-1)*(n+2)/2).

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

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A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

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

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A014431 a(1) = 1, a(2) = 2, a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

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Magma

Mathematica

PARI

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Extensions

A110180 Triangle of generalized central trinomial coefficients.

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Mathematica

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

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).

A014431 a(1) = 1, a(2) = 2, a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)C(j,n-j)2^(n-j).