A122868 -id:A122868 - cp's OEIS frontend

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).

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).

A220178 Triangle where the g.f. for row n equals d^n/dx^n (1+x+x^2)^n / n! for n>=0, as read by rows.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 7, 24, 30, 20, 19, 80, 150, 140, 70, 51, 270, 630, 840, 630, 252, 141, 882, 2520, 4200, 4410, 2772, 924, 393, 2856, 9576, 19320, 25410, 22176, 12012, 3432, 1107, 9144, 35280, 83160, 131670, 144144, 108108, 51480, 12870, 3139, 29070, 126720, 341880, 630630, 828828, 780780, 514800, 218790, 48620

Offset: 0

Paul D. Hanna, Dec 06 2012

			Triangle begins:
1;
1, 2;
3, 6, 6;
7, 24, 30, 20;
19, 80, 150, 140, 70;
51, 270, 630, 840, 630, 252;
141, 882, 2520, 4200, 4410, 2772, 924;
393, 2856, 9576, 19320, 25410, 22176, 12012, 3432;
1107, 9144, 35280, 83160, 131670, 144144, 108108, 51480, 12870; ...
The g.f. for column k>=0 equals the central binomial coefficient C(2*k,k) times x^k*y^k*G(x)^(2*k+1) where G(x) = 1/sqrt(1-2*x-3*x^2) is the g.f. of the central trinomial coefficients A002426.
The g.f. for row n is d^n/dx^n (1+x+x^2)^n/n!, which begins:
n=0: 1;
n=1: 1 + 2*x;
n=2: 3 + 6*x + 6*x^2;
n=3: 7 + 24*x + 30*x^2 + 20*x^3;
n=4: 19 + 80*x + 150*x^2 + 140*x^3 + 70*x^4;
n=5: 51 + 270*x + 630*x^2 + 840*x^3 + 630*x^4 + 252*x^5;
n=6: 141 + 882*x + 2520*x^2 + 4200*x^3 + 4410*x^4 + 2772*x^5 + 924*x^6; ...

Ivan Neretin, Table of n, a(n) for n = 0..5150 (rows 0..100)

Cf. A002426 (first column), A000984 (main diagonal), A122868 (row sums).

Flatten@Table[CoefficientList[D[(1 + x + x^2)^n/n!, {x, n}], x], {n, 0, 9}] (* Ivan Neretin, Jun 22 2019 *)

{T(n,k)=polcoeff(polcoeff(1/sqrt(1-2*x-3*x^2 - 4*x*y +x*O(x^n)+y*O(y^k)),n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

row(n) = my(p=(1+x+x^2)^n / n!); for (k=1, n, p = deriv(p)); Vecrev(p); \\ Michel Marcus, Jun 22 2019

G.f.: A(x,y) = 1 / sqrt(1-2*x-3*x^2 - 4*x*y).
G.f.: A(x,y) = Sum_{k>=0} binomial(2*k,k) * x^k*y^k / (1-2*x-3*x^2)^(k+1/2).
First column is the central trinomial coefficients (A002426).
Main diagonal is the central binomial coefficients (A000984).
Row sums form the central coefficients of (1+3*x+3*x^2)^n (A122868).

A225439 Expansion of 3x/((1-(1-9x)^(1/3))(1-9x)^(2/3)).

Original entry on oeis.org

1, 3, 21, 162, 1305, 10773, 90342, 765936, 6546177, 56293380, 486451251, 4220183916, 36731240910, 320571837810, 2804298945840, 24580601689752, 215832643307217, 1898042178972285, 16714070686567620, 147360883148636850, 1300623629653125855

Offset: 0

Vladimir Kruchinin, May 08 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A025748, A097188, A122868.

A225439 := n -> `if`(n=0,1,(GAMMA(n+2/3)/GAMMA(2/3)+GAMMA(n+1/3)/(GAMMA(1/3)))* 3^(2*n-1)/GAMMA(n+1)): seq(A225439(i),i=0..20); # Peter Luschny, Jul 05 2013

Table[Sum[Binomial[k,n-k]*3^k*(-1)^(n-k)*Binomial[n+k-1,n-1], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 22 2013 *)

a(n):=if n=0 then 1 else sum(binomial(k,n-k)*3^(k)*(-1)^(n-k)*binomial(n+k-1,n-1),k,0,n);

my(x='x+O('x^66)); Vec(3*x/((1-(1-9*x)^(1/3))*(1-9*x)^(2/3))) \\ Joerg Arndt, May 08 2013

{a(n)=local(B=(1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x));polcoeff(1+x*B'/B, n, x)} \\ Paul D. Hanna, May 08 2013

a(n) = Sum_{k = 0..n} C(k,n-k)*3^(k)*(-1)^(n-k)*C(n+k-1,n-1), n>0, a(0)=1.
G.f.: A(x) = 1 + x*B'(x)/B(x), where B(x) = (1-(1-9*x)^(1/3))/(3*x) is the g.f. of A097188.
n*(n-1)*a(n) = 18*(n-1)^2*a(n-1) - 9*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, May 22 2013
a(n) ~ 3^(2*n-1)/(GAMMA(2/3)*n^(1/3)). - Vaclav Kotesovec, May 22 2013
a(n) = ((Gamma(n+2/3)/Gamma(2/3))+(Gamma(n+1/3)/Gamma(1/3)))*3^(2*n-1)/Gamma(n+1) for n > 0. - Peter Luschny, Jul 05 2013
From Peter Bala, Mar 11 2022: (Start)
a(n) = [x^n] (1/(1 - 3*x + 3*x^2))^n. Cf. A122868(n) = [x^n] (1 + 3*x + 3*x^2)^n.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

A232973 Dziemianczuk's array S(i,j) read by antidiagonals.

Original entry on oeis.org

1, 3, 6, 15, 33, 60, 81, 189, 378, 675, 459, 1107, 2349, 4509, 7992, 2673, 6588, 14553, 29403, 55188, 97416, 15849, 39663, 90207, 189351, 371358, 687258, 1209951, 95175, 240894, 560115, 1211031, 2458998, 4727565, 8664813, 15227190, 576963, 1473147, 3485187

Offset: 0

N. J. A. Sloane, Dec 05 2013

			Array begins:
     1,     3,     15,      81,      459,      2673,     15849, ...
     6,    33,    189,    1107,     6588,     39663,    240894, ...
    60,   378,   2349,   14553,    90207,    560115,   3485187, ...
   675,  4509,  29403,  189351,  1211031,   7715331,  49045662, ...
  7992, 55188, 371358, 2458998, 16112925, 104838219, 678790125, ...
  ...

Lars Blomberg, Table of n, a(n) for n = 0..5049 (The first 100 antidiagonals)
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.

See A122868 and A232969 for leading row row and column.

\\ Dziemianczuk, Proposition 1
S(n,k)=sum(i=0,n+k,sum(j=0,i,binomial(k,j)*binomial(j,i-j)*binomial(2*k+n-i,k)));
A=[]; for(i=1,10,A=concat(A,vector(i,j,S(j-1,i-1))));
A \\ Lars Blomberg, Jul 20 2017

a(15)-a(38) from Lars Blomberg, Jul 20 2017

A260774 Certain directed lattice paths.

Original entry on oeis.org

1, 6, 33, 189, 1107, 6588, 39663, 240894, 1473147, 9058554, 55954395, 346934745, 2157989445, 13459891500, 84152389833, 527224251861, 3309194474451, 20804569738218, 130987600581699, 825796890644895, 5212349717906889, 32935490120006604, 208316726580941037

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Alois P. Heinz, Table of n, a(n) for n = 0..1235 (first 101 terms from Lars Blomberg)
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

Cf. A122868, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 21 2021

b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
     If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
     If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

See Dziemianczuk (2014) Equation (33a) with m=1.
From Vaclav Kotesovec, Jul 15 2022: (Start)
Recurrence: (n+1)*(4*n - 3)*a(n) = 6*(4*n^2 - n - 1)*a(n-1) + 3*(n-1)*(4*n + 1)*a(n-2).
a(n) ~ (3 + 2*sqrt(3))^(n+1) / sqrt(6*Pi*n). (End)

More terms from Lars Blomberg, Aug 01 2015

A243116 a(n) = Sum_{k=0..n} C(n + 2k, 3k) * C(3k, 2k).

Original entry on oeis.org

1, 4, 28, 220, 1816, 15424, 133456, 1169872, 10354528, 92331904, 828204928, 7464652672, 67547774464, 613295870464, 5584367987968, 50974595472640, 466307503244800, 4273832891668480, 39237007284226048, 360768875975526400, 3321625537178669056, 30619908430235828224, 282578914501599305728

Offset: 0

Paul D. Hanna, Aug 20 2014

nonn

Compare to: Sum_{k=0..n} (-1)^k * C(n+2*k,3*k) * C(3*k,2*k) = (-2)^n for n>=0.

			G.f.: A(x) = 1 + 4*x + 28*x^2 + 220*x^3 + 1816*x^4 + 15424*x^5 +...
where
A(x) = 1/(1-x) + 3*x/(1-x)^4 + 15*x^2/(1-x)^7 + 84*x^3/(1-x)^10 + 495*x^4/(1-x)^13 + 3003*x^5/(1-x)^16 + 18564*x^6/(1-x)^19 + 116280*x^7/(1-x)^22 + 735471*x^8/(1-x)^25 +...+ C(3*n, n)*x^n/(1-x)^(3*n+1) +...
ILLUSTRATION OF TERMS.
The sequence A005809(k) = C(3*k, 2*k) begins:
  [1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, ...];
the triangle A109955(n,k) = C(n + 2*k, 3*k) begins:
  1;
  1, 1;
  1, 4, 1;
  1, 10, 7, 1;
  1, 20, 28, 10, 1;
  1, 35, 84, 55, 13, 1;
  1, 56, 210, 220, 91, 16, 1;
  1, 84, 462, 715, 455, 136, 19, 1; ...
where a(n) = Sum_{k=0..n} A109955(n,k) * A005809(k):
  a(1) = 1*1 + 1*3 = 4;
  a(2) = 1*1 + 4*3 + 1*15 = 28;
  a(3) = 1*1 + 10*3 + 7*15 + 1*84 = 220;
  a(4) = 1*1 + 20*3 + 28*15 + 10*84 + 1*495 = 1816; ...
compare to: Sum_{k=0..n} (-1)^k * A109955(n,k) * A005809(k) = (-2)^n.

Michael De Vlieger, Table of n, a(n) for n = 0..1027
Hacène Belbachir and Abdelghani Mehdaoui, Diagonal sums in Pascal pyramid (1, 2, r), Les Annales RECITS (2019) Vol. 6, 45-52.

Cf. A109955, A005809.

Table[Sum[Binomial[n + 2*k, 3*k] * Binomial[3*k, 2*k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 21 2014 *)

{a(n)=sum(k=0,n, binomial(n+2*k,3*k) * binomial(3*k,2*k))}
for(n=0,30,print1(a(n),", "))

{a(n)=-(-2)^n + 2*sum(k=0, n\2, binomial(n+4*k, 6*k) * binomial(6*k, 4*k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(A=1); A=sum(m=0, n, binomial(3*m, m) * x^m/(1-x +x*O(x^n))^(3*m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} C(3*n, n) * x^n / (1-x)^(3*n+1).  - Paul D. Hanna, Aug 30 2014
G.f.: 1/(1-x) / ( 3 / G(x/(1-x)^3) - 2 ), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764. - Paul D. Hanna, Aug 30 2014
G.f. satisfies: A(x) = 1 + (4-3*x)*A(x) - (4 - 39*x + 12*x^2 - 4*x^3)*A(x)^3. - Paul D. Hanna, Sep 05 2014
a(n) = Sum_{k=0..n} A109955(n,k) * A005809(k).
a(n) = -(-2)^n + 2*Sum_{k=0..[n/2]} C(n+4*k, 6*k) * C(6*k, 4*k).
Recurrence: 2*n*(2*n-1)*(3*n-4)*a(n) = (3*n-2)*(39*n^2 - 65*n + 18)*a(n-1) - 2*(n-1)*(18*n^2 - 33*n + 10)*a(n-2) + 4*(n-2)*(n-1)*(3*n-1)*a(n-3). - Vaclav Kotesovec, Aug 21 2014
From Peter Bala, Mar 11 2022: (Start)
a(n) = Sum_{k = 0..floor(n/4)} (-1)^k*binomial(n,k)*binomial(4*n-4*k,3*n).
a(n) = [x^n] ( (1 + x)^4 - x^4 )^n. Cf. A122868(n) = [x^n] ( (1 + x)^3 - x^3 )^n.
It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
a(n) = [x^n] (1 + x + (1 + x)^3)^n. - Seiichi Manyama, Nov 25 2024

A260772 Certain directed lattice paths.

Original entry on oeis.org

1, 3, 10, 41, 190, 946, 4940, 26693, 147990, 837102, 4811860, 28027210, 165057100, 981177060, 5879570200, 35478788269, 215398416870, 1314794380374, 8064119033220, 49673222082782, 307163049317540, 1906066361809148, 11865666767361960, 74081851132379426

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Lars Blomberg, Table of n, a(n) for n = 0..100
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui, Solving Third Order Linear Difference Equations in Terms of Second Order Equations, arXiv:2402.11121 [math.AC], 2024. See p. 3.

Cf. A122868, A260774, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

# A260772 satisfies a 4th-order recurrence that can be reduced
# to a 2nd-order recurrence given in this program t:
t := proc(n) options remember;
if n <= 1 then
    [-1/2, 0, 1, 4][2*n+2]
  else
    (16*(n-2)*(2*n-3)*(5*n-2)*t(n-2) + (440*n^3-1056*n^2+724*n-144)*t(n-1))
       /( n*(2*n+1)*(5*n-7) )
  fi
end:
A260772 := proc(n)
t(n/2) + ( (2-2*n)*t((n-1)/2)+(n+2)*t((n+1)/2) ) / (1+5*n)
end:
seq(A260772(i),i=0..100);
# Mark van Hoeij, Jul 14 2022

a(n):=if n=0 then 1 else sum((-1)^j*binomial(n,j)*binomial(3*n-4*j,n-4*j+1),j,0,(n+1)/4)/n; /* Vladimir Kruchinin, Apr 04 2019 */

a(n) = if (n==0, 1, sum(j=0, (n+1)/4, (-1)^j*binomial(n,j)*binomial(3*n-4*j, n-4*j+1))/n); \\ Michel Marcus, Apr 05 2019

G.f.: P1(x) = (2*(1-x)/3)/x - ((2*sqrt(1-5*x-2*x^2)/3)/x)*sin((Pi/6 + arccos(((20*x^3-6*x^2+15*x-2)/2)/(1-5*x-2*x^2)^(3/2))/3)). - See Dziemianczuk (2014), Proposition 11.
a(n) = (1/n)*Sum_{j=0..(n+1)/4} (-1)^j*C(n,j)*C(3*n-4*j,n-4*j+1), a(0)=1. - Vladimir Kruchinin, Apr 04 2019
n*(n+1)*(25*n^2-70*n+21)*a(n) - 30*(7*n-15)*n*a(n-1) + (-1100*n^4+5280*n^3-6424*n^2-1188*n+3816)*a(n-2) + 120*(n+2)*(n-3)*a(n-3) - 16*(n-3)*(n-4)*(25*n^2-20*n-24)*a(n-4) = 0. - Mark van Hoeij, Jul 14 2022
a(n) ~ 2^(n - 1/2) * phi^((10*n - 1)/4) / (sqrt(Pi) * 5^(1/4) * sqrt(phi^(3/2) - 2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 15 2022

More terms from Lars Blomberg, Aug 01 2015

cp's OEIS Frontend

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

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A220178 Triangle where the g.f. for row n equals d^n/dx^n (1+x+x^2)^n / n! for n>=0, as read by rows.

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A225439 Expansion of 3*x/((1-(1-9*x)^(1/3))*(1-9*x)^(2/3)).

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A232973 Dziemianczuk's array S(i,j) read by antidiagonals.

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A260774 Certain directed lattice paths.

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Maple

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A243116 a(n) = Sum_{k=0..n} C(n + 2*k, 3*k) * C(3*k, 2*k).

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A260772 Certain directed lattice paths.

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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).

A225439 Expansion of 3x/((1-(1-9x)^(1/3))(1-9x)^(2/3)).

A243116 a(n) = Sum_{k=0..n} C(n + 2k, 3k) * C(3k, 2k).