A098479 -id:A098479 - cp's OEIS frontend

A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

1, 1, 0, 1, 2, 0, 1, 6, 0, 0, 1, 12, 6, 0, 0, 1, 20, 30, 0, 0, 0, 1, 30, 90, 20, 0, 0, 0, 1, 42, 210, 140, 0, 0, 0, 0, 1, 56, 420, 560, 70, 0, 0, 0, 0, 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0, 1, 90, 1260, 4200, 3150, 252, 0, 0, 0, 0, 0, 1, 110, 1980, 9240, 11550, 2772, 0, 0, 0, 0, 0, 0, 1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0, 1, 156, 4290, 34320, 90090, 72072, 12012, 0, 0, 0, 0, 0, 0, 0, 1, 182, 6006, 60060, 210210, 252252, 84084, 3432, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 31 2003

The rows of this triangle are the gamma vectors of the n-dimensional type B associahedra (Postnikov et al., p.38 ). Cf. A055151 and A101280. - Peter Bala, Oct 28 2008
T(n,k) is the number of Grand Motzkin paths of length n having exactly k upsteps (1,1). Cf. A109189, A055151. - Geoffrey Critzer, Feb 05 2014
The result Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*x^k = (sqrt(1 - 4*x))^n* P(n,1/sqrt(1 - 4*x)) expressing the row polynomials of this triangle in terms of the Legendre polynomials P(n,x) is due to Catalan. See Laden, equation 7.10, p. 56. - Peter Bala, Mar 18 2018

			Triangle begins:
  1
  1,   0
  1,   2,    0
  1,   6,    0,     0
  1,  12,    6,     0,     0
  1,  20,   30,     0,     0,     0
  1,  30,   90,    20,     0,     0,   0
  1,  42,  210,   140,     0,     0,   0, 0
  1,  56,  420,   560,    70,     0,   0, 0, 0
  1,  72,  756,  1680,   630,     0,   0, 0, 0, 0
  1,  90, 1260,  4200,  3150,   252,   0, 0, 0, 0, 0
  1, 110, 1980,  9240, 11550,  2772,   0, 0, 0, 0, 0, 0
  1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0
Relocating the zeros to be evenly distributed and interpreting the triangle as the coefficients of polynomials
                     1
                     1
                 1 + 2 q^2
                 1 + 6 q^2
            1 + 12 q^2 +  6 q^4
            1 + 20 q^2 + 30 q^4
       1 + 30 q^2 +  90 q^4 +  20 q^6
       1 + 42 q^2 + 210 q^4 + 140 q^6
  1 + 56 q^2 + 420 q^4 + 560 q^6 + 70 q^8
then the substitution q^k -> 1/(floor(k/2)+1) gives the Motzkin numbers A001006.
- _Peter Luschny_, Aug 29 2011

Alois P. Heinz, Rows n = 0..140, flattened
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.
Hyman N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007.

Row sums A002426. Antidiagonal sums A098479.

Cf. A055151, A063007, A109187, A101280.

for i from 0 to 12 do seq(binomial(i, j)*binomial(i-j, j), j=0..i) od; # Zerinvary Lajos, Jun 07 2006
# Alternatively:
R := (n, x) -> simplify(hypergeom([1/2 - n/2, -n/2], [1], 4*x)):
Trow := n -> seq(coeff(R(n,x), x, j), j=0..n):
seq(print(Trow(n)), n=0..9); # Peter Luschny, Mar 18 2018

nn=15;mxy=(1-x-(1-2x+x^2-4x^2y)^(1/2))/(2x^2 y);Map[Select[#,#>0&]&, CoefficientList[Series[1/(1-x-2y x^2mxy),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 05 2014 *)

T(n,k) = binomial(n,2*k)*binomial(2*k,k);
concat(vector(15, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 01 2018

T(n,k) = n!/((n-2*k)!*k!*k!).
E.g.f.: exp(x)*BesselI(0, 2*x*sqrt(y)). - Vladeta Jovovic, Apr 07 2005
O.g.f.: ( 1 - x - sqrt(1 - 2*x + x^2 - 4*x^2*y))/(2*x^2*y). - Geoffrey Critzer, Feb 05 2014
R(n, x) = hypergeom([1/2 - n/2, -n/2], [1], 4*x) are the row polynomials. - Peter Luschny, Mar 18 2018
From Peter Bala, Jun 23 2023: (Start)
T(n,k) = Sum_{i = 0..k} (-1)^i*binomial(n, i)*binomial(n-i, k-i)^2. Cf. A063007(n,k) = Sum_{i = 0..k} binomial(n, i)^2*binomial(n-i, k-i).
T(n,k) = A063007(n-k,k); that is, the diagonals of this table are the rows of A063007. (End)

A098551 Inverse of A098550.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 15, 6, 5, 16, 22, 12, 23, 8, 7, 14, 30, 31, 43, 18, 17, 20, 51, 33, 11, 25, 19, 27, 61, 39, 62, 29, 24, 35, 13, 37, 79, 41, 21, 48, 87, 44, 88, 46, 26, 56, 101, 52, 40, 50, 28, 54, 114, 69, 34, 58, 47, 63, 127, 71, 132, 60, 42, 65, 36, 73, 142, 67, 49, 80, 153

Offset: 1

Reinhard Zumkeller, Sep 14 2004

nonn

Now known to be a permutation of the natural numbers: see the 2015 article by Applegate, Havermann, Selcoe, Shevelev, Sloane, and Zumkeller.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015.
Index entries for sequences that are permutations of the natural numbers

Cf. A098550, A098553.
Cf. A249943 (partial maxima).
Cf. A098479, A255940.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a098551 = (+ 1) . fromJust . (`elemIndex` a098550_list)
-- Reinhard Zumkeller, Nov 21 2014

f[lst_List] := Block[{k = 4}, While[ GCD[ lst[[-2]], k] == 1 || GCD[ lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]]; Table[ Position[ Nest[ f, {1, 2, 3}, 120], n], {n, 71}] // Flatten (* Robert G. Wilson v, Nov 21 2014 *)

A098553(n) = a(a(n)).

A376791 Expansion of 1/sqrt((1 - x^3)^2 - 4*x).

Original entry on oeis.org

1, 2, 6, 21, 76, 282, 1065, 4074, 15732, 61193, 239406, 941064, 3713701, 14703896, 58383138, 232383841, 926943678, 3704410890, 14828984641, 59450138412, 238659074286, 959247218253, 3859777477944, 15546444564846, 62675854384977, 252893414725842, 1021208266423260

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

From Seiichi Manyama, Apr 30 2025: (Start)
Number of lattice paths from (0,0) to (n,n) using steps (1,0),(0,1),(3,3).
Diagonal of the rational function 1 / (1 - x - y - x^3*y^3). (End)

Cf. A001850, A349713, A376792.

Cf. A053442, A098479.

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x^3)^2-4*x))

a(n) = sum(k=0, n\3, binomial(2*n-5*k, k)*binomial(2*n-6*k, n-3*k));

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k,k) * binomial(2*n-6*k,n-3*k).

A098482 Expansion of 1/sqrt((1-x)^2 - 4*x^4).

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 13, 21, 37, 73, 147, 283, 531, 1007, 1953, 3817, 7423, 14371, 27877, 54333, 106189, 207585, 405743, 793719, 1554889, 3049525, 5984803, 11751067, 23086695, 45388291, 89289765, 175746797, 346077153, 681795925, 1343790319, 2649687079, 5226711507

Offset: 0

Paul Barry, Sep 10 2004

From Joerg Arndt, Jul 01 2011: (Start)
Empirical: Number of lattice paths from (0,0) to (n,n) using steps (4,0), (0,4), (1,1).
It appears that 1/sqrt((1-x)^2-4*x^s) is the g.f. for lattice paths from (0,0) to (n,n) using steps (s,0), (0,s), (1,1).
Empirical: Number of lattice paths from (0,0) to (n,n) using steps (3,1), (1,3), (1,1).  (End)
1/sqrt((1-x)^2-4*r*x^4) expands to sum(k=0..floor(n/2), binomial(n-2*k,k)*binomial(n-3*k,k)*r^k ).
Diagonal of the rational function 1 / ((1-x)*(1-y) - x^3*y^4). - Seiichi Manyama, Apr 29 2025
a(n) is the number of ways to tile a strip of length n with 1 X 1 squares and 1 X 2 red dominos and 1 X 2 blue dominos, with an equal number of red and blue dominos. - Greg Dresden and Leo Zhang, Jul 08 2025

			From _Joerg Arndt_, Jul 01 2011: (Start)
The triangle of lattice paths from (0,0) to (n,k) using steps (3,1), (1,3), (1,1) begins
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 2, 0, 3;
  0, 0, 0, 3, 0, 7;
  0, 0, 1, 0, 4, 0, 13;
  0, 0, 0, 3, 0, 8, 0, 21;
  0, 0, 0, 0, 6, 0, 18, 0, 37;
  0, 0, 0, 1, 0, 10, 0, 37, 0, 73;
The triangle of lattice paths from (0,0) to (n,k) using steps (4,0), (0,4), (1,1) begins
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  1, 0, 0, 0, 3;
  0, 2, 0, 0, 0, 7;
  0, 0, 3, 0, 0, 0, 13;
  0, 0, 0, 4, 0, 0, 0, 21;
  1, 0, 0, 0, 8, 0, 0, 0, 37;
  0, 3, 0, 0, 0, 18, 0, 0, 0, 73;
The diagonals of both appear to be this sequence.  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

Cf. A098479, A098483, A098484.

seq(add(binomial(n-3*k,k)*binomial(n-2*k,k),k=0..floor(n/3)),n=0..34); # Zerinvary Lajos, Apr 03 2007

CoefficientList[Series[1/Sqrt[(1-x)^2-4*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

/* as lattice paths, assuming the first comment is true */
/* same as in A092566 but use either of */
steps=[[4,0], [0,4], [1,1]];
steps=[[3,1], [1,3], [1,1]];
/* Joerg Arndt, Jul 01 2011 */

a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k, k) * binomial(n-3*k, k).
D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 4*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014
a(n) ~ 2^(n+1/2) / sqrt(3*Pi*n). - Vaclav Kotesovec, Jun 23 2014
G.f.: 1/(1 - x - 2*x^4/(1 - x - x^4/(1 - x - x^4/(1 - x - x^4/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021
a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k, 2*k) * binomial(2*k, k). - Greg Dresden and Leo Zhang, Jul 08 2025

A098480 Expansion of 1/sqrt((1-x)^2-8x^3).

Original entry on oeis.org

1, 1, 1, 5, 13, 25, 65, 181, 445, 1113, 2945, 7685, 19821, 51865, 136513, 358229, 942109, 2487385, 6573825, 17387045, 46066253, 122213913, 324512833, 862511605, 2294698109, 6109933657, 16280439937, 43411979845, 115835462445

Offset: 0

Paul Barry, Sep 10 2004

1/sqrt((1-x)^2-4rx^3) expands to sum{k=0..floor(n/2), binomial(n-k,k)binomial(n-2k,k)r^k}

Michael De Vlieger, Table of n, a(n) for n = 0..2309
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Cf. A098479, A098481.

Array[Sum[Binomial[# - k, k] Binomial[# - 2 k, k] 2^k, {k, 0, #/2}] &, 29, 0] (* Michael De Vlieger, Jul 16 2019 *)

a(n)=sum{k=0..floor(n/2), binomial(n-k, k)binomial(n-2k, k)2^k}. D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +(n-1)*a(n-2) +4*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Nov 10 2014

A098481 Expansion of 1/sqrt((1-x)^2 - 12*x^3).

Original entry on oeis.org

1, 1, 1, 7, 19, 37, 115, 361, 937, 2599, 7777, 22195, 62701, 182647, 531829, 1534903, 4461571, 13034917, 38015899, 110994193, 325011151, 952442557, 2792471239, 8198275933, 24093817531, 70852613041, 208516575043, 614145137137

Offset: 0

Paul Barry, Sep 10 2004

1/sqrt((1-x)^2 - 4*r*x^3) expands to Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n-2k, k)*r^k.

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 0..200 from Vincenzo Librandi)

Cf. A098479, A098480.

CoefficientList[Series[1/Sqrt[(1-x)^2-12x^3],{x,0,40}],x] (* Harvey P. Dale, Jun 02 2011 *)

Vec(1/sqrt((1-x)^2 - 12*x^3) + O(x^50)) \\ G. C. Greubel, Jan 30 2017

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n-2k, k)*3^k.
D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 6*(2*n-3)*a(n-3). - Vaclav Kotesovec, Jun 23 2014
a(n) ~ 3^(n+1) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 23 2014

A120580 Hankel transform of Sum_{k=0..n} C(2k,k).

Original entry on oeis.org

1, 0, -4, -8, 0, 32, 64, 0, -256, -512, 0, 2048, 4096, 0, -16384, -32768, 0, 131072, 262144, 0, -1048576, -2097152, 0, 8388608, 16777216, 0, -67108864, -134217728, 0, 536870912, 1073741824, 0, -4294967296, -8589934592, 0, 34359738368, 68719476736, 0, -274877906944, -549755813888, 0

Offset: 0

Paul Barry, Jun 15 2006

Hankel transform of A006134.
Hankel transform of A098479. - Paul Barry, Sep 19 2008
Hankel transform of A025565. - Paul Barry, Mar 26 2010

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A006134, A025565, A098479, A104538.

LinearRecurrence[{2,-4},{1,0},50] (* Harvey P. Dale, Feb 13 2023 *)

G.f.: (1-2x)/(1-2x+4x^2).
a(n) = 2^n*(cos(Pi*n/3)-sin(Pi*n/3)/sqrt(3)).
E.g.f.: exp(x)*(cos(sqrt(3)*x) - sin(sqrt(3)*x)/sqrt(3)). - Stefano Spezia, Jul 15 2024

A113179 Expansion of 1/sqrt((1-2x)^2-8x^3).

Original entry on oeis.org

1, 2, 4, 12, 40, 128, 408, 1328, 4384, 14560, 48576, 162816, 547936, 1850048, 6263680, 21257856, 72298240, 246345728, 840775424, 2873802240, 9835840512, 33704557568, 115622041600, 397032488960, 1364610270720, 4694145256448

Offset: 0

Paul Barry, Oct 16 2005

In general, 1/sqrt((1-a*x)^2-4*b*x^3) expands to sum{k=0..floor(n/2), C(n-k,k)C(n-2k,k)b^k*a^(n-3k)}.

Michael De Vlieger, Table of n, a(n) for n = 0..1837
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Cf. A098479.

CoefficientList[Series[1/Sqrt[(1-2x)^2-8x^3],{x,0,30}],x] (* Harvey P. Dale, Dec 23 2017 *)

a(n)=sum{k=0..floor(n/2), C(n-k, k)C(n-2k, k)2^(n-2k)}.

D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +4*(n-1)*a(n-2) +4*(-2*n+3)*a(n-3)=0. [Belbachir]

A383539 a(n) = Sum_{k=0..floor(n/3)} binomial(n-k,k)^2 * binomial(n-2*k,k).

Original entry on oeis.org

1, 1, 1, 5, 19, 49, 137, 481, 1645, 5259, 17309, 59477, 203931, 693865, 2384149, 8277773, 28797631, 100312525, 350891175, 1232122535, 4335809699, 15287669469, 54029225569, 191351513905, 678850904981, 2412164275651, 8584573648693, 30595269827149

Offset: 0

Seiichi Manyama, Apr 29 2025

nonn

Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - x^2*y^2*z^3).

Robert Israel, Table of n, a(n) for n = 0..1758

Cf. A383525, A383537.

Cf. A098479.

f:= proc(n) local k; add(binomial(n-k,k)^2 * binomial(n-2*k,k),k=0..n/3) end proc:
map(f, [$0..30]); # Robert Israel, May 29 2025

a(n) = sum(k=0, n\3, binomial(n-k, k)^2*binomial(n-2*k, k));

A182885 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2. These are paths that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1; an (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 7, 3, 1, 13, 10, 3, 27, 29, 6, 1, 61, 66, 22, 4, 133, 157, 75, 10, 1, 287, 398, 201, 40, 5, 633, 975, 538, 155, 15, 1, 1407, 2334, 1506, 476, 65, 6, 3121, 5631, 4077, 1414, 280, 21, 1, 6943, 13602, 10695, 4320, 966, 98, 7, 15517, 32623, 27966, 12765, 3150, 462, 28, 1

Offset: 0

Emeric Deutsch, Dec 11 2010

Sum of entries in row n is A051286(n).
T(n,0)=A098479(n).
Sum(k*T(n,k), k=0..n)=A182886(n).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them have exactly one H step.
Triangle starts:
1;
1;
1,1;
3,2;
7,3,1;
13,10,3;
27,29,6,1;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A098479, A182886.

G:=1/sqrt(1-2*z-2*t*z^2+z^2+2*t*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form

G.f.: G(t,z) =1/sqrt(1-2z-2tz^2+z^2+2t*z^3+t^2*z^4-4z^3).

cp's OEIS Frontend

A089627 T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

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A098551 Inverse of A098550.

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A376791 Expansion of 1/sqrt((1 - x^3)^2 - 4*x).

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A098482 Expansion of 1/sqrt((1-x)^2 - 4*x^4).

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A098480 Expansion of 1/sqrt((1-x)^2-8x^3).

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Mathematica

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A098481 Expansion of 1/sqrt((1-x)^2 - 12*x^3).

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Mathematica

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A120580 Hankel transform of Sum_{k=0..n} C(2k,k).

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Mathematica

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A113179 Expansion of 1/sqrt((1-2x)^2-8x^3).

A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.