A026641 -id:A026641 - cp's OEIS frontend

A382848 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(n+k,k).

1, 1, -5, -35, -29, 751, 3991, -4115, -137885, -495269, 2114245, 25786795, 50109775, -627370925, -4643568305, -495798035, 157753390435, 768269873875, -1851203127335, -35924154988865, -107001450483779, 763444753890721, 7510024190977105, 8899910747771995

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

sign

Diagonal of the rational function 1 / (1 + x + x*y + y*z + x*z + x*y*z).

Cf. A005258, A026641, A126869, A245086, A382405, A382849.

Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[n + k, k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, -1], {n, 0, 23}]
Table[SeriesCoefficient[1/(1 + x + x y + y z + x z + x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 23}]

(59*n-94)*n^2*a(n) = 5*(59*n^3-153*n^2+117*n-30)*a(n-1) - (2301*n^3-8268*n^2+9257*n-3050)*a(n-2) - 2*(59*n-35)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = -5. - Peter Bala, May 24 2025

A188289 Binomial sum related to rooted trees.

Original entry on oeis.org

0, 2, 3, 14, 45, 167, 609, 2270, 8517, 32207, 122463, 467843, 1794195, 6903353, 26635773, 103020254, 399300165, 1550554583, 6031074183, 23493410759, 91638191235, 357874310213, 1399137067683, 5475504511859, 21447950506395, 84083979575117

Offset: 0

Olivier Gérard, Aug 19 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000984, A014300, A026641, A178792, A176479, A072547.

List([0..30], n-> Binomial(2*n,n) -(-1)^n -Sum([0..n-1], k-> Binomial(2*k,n-1))); # G. C. Greubel, Apr 29 2019

[n eq 0 select 0 else Binomial(2*n, n) -(-1)^n - (&+[Binomial(2*k, n-1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Apr 29 2019

Table[Binomial[2n,n]-(-1)^n-Sum[Binomial[2k,n-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Dec 10 2012 *)

{a(n) = binomial(2*n,n) -(-1)^n -sum(k=0,n-1, binomial(2*k,n-1))}; \\ G. C. Greubel, Apr 29 2019

[binomial(2*n,n) -(-1)^n -sum(binomial(2*k, n-1) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Apr 29 2019

a(n) = binomial(2*n,n) - (-1)^n - Sum_{k=0..n-1} binomial(2*k, n-1).
a(n) = Sum_{k=1..n} binomial(n+k,k)*(Sum_{r=n-k..n} (-1)^r*binomial(n-k, r)).
a(n) = (-1)^n*2^(-(1+n))*(1 - 2^(1+n) + (-2)^n*binomial(2+2*n, 1+n) * hypergeometric2F1(1, 2+2*n; 2+n; -1)).
a(n) = Sum_{k=1..n} (-1)^(n+k)*binomial(n+k,k). - Ridouane Oudra, Sep 07 2025

A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).

Original entry on oeis.org

1, 3, 14, 71, 379, 2082, 11651, 66051, 378064, 2180037, 12644861, 73695358, 431209313, 2531556197, 14904832196, 87970766447, 520337606401, 3083584244460, 18304476242735, 108820740004749, 647817646760368, 3861215365595659, 23039691494489015, 137615812845579390

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 9, 29.

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^4+2*x^3-x^2-6*x+1) )); // G. C. Greubel, Apr 29 2019

CoefficientList[Series[1/Sqrt[x^4 + 2 x^3 - x^2 - 6 x + 1], {x, 0, 23}], x] (* Michael De Vlieger, Oct 08 2016 *)

/* same as in A092566 but use */
steps=[[0,1], [1,0], [1,1], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

my(x='x+O('x^30)); Vec(1/sqrt(x^4+2*x^3-x^2-6*x+1)) \\ G. C. Greubel, Apr 29 2019

(1/sqrt(x^4+2*x^3-x^2-6*x+1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

G.f.: 1/sqrt(x^4 +2*x^3 -x^2 -6*x +1). - Mark van Hoeij, Apr 17 2013

D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +(-n+1)*a(n-2) +(2*n-3)*a(n-3) +(n-2)*a(n-4)=0. - R. J. Mathar, Oct 08 2016

A192371 Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).

Original entry on oeis.org

1, 1, 3, 9, 25, 87, 307, 1113, 4149, 15605, 59201, 225999, 866449, 3333847, 12865335, 49769689, 192945411, 749396493, 2915432049, 11358771965, 44313108627, 173081422997, 676766482917, 2648843996031, 10376891445525, 40685535827325, 159641884780749, 626849029013919, 2463010645910537, 9683604464279235

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.

s := RootOf( (s^3-s-1)*(s-1)+x*s*(4-3*s), s);
ogf := sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)):
series(ogf, x=0, 30);  # Mark van Hoeij, Apr 17 2013
# second Maple program:
b:= proc(p) b(p):= `if`(p=[0$2], 1, `if`(min(p[])<0, 0,
      add(b(p-l), l=[[1, 1], [0, 2], [2, 0], [0, 3], [3, 0]])))
    end:
a:= n-> b([n$2]):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 18 2014

b[p_List] := b[p] = If[p == {0, 0}, 1, If[Min[p] < 0, 0, Sum[b[p - l], {l, {{1, 1}, {0, 2}, {2, 0}, {3, 0}, {0, 3}}}]]]; a[n_] := b[{n, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

/* same as in A092566 but use */
steps=[[1,1], [2,0], [0,2], [3,0], [0,3]];
/* Joerg Arndt, Jun 30 2011 */

G.f.: sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)) where the function s satisfies (s^3-s-1)*(s-1)+x*s*(4-3*s) = 0. - Mark van Hoeij, Apr 17 2013

A192417 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,2), (3,3).

Original entry on oeis.org

1, 2, 7, 27, 107, 436, 1810, 7609, 32288, 138009, 593311, 2562725, 11112720, 48347332, 210936119, 922550622, 4043488129, 17755735241, 78099099877, 344033901804, 1517535718392, 6701979806379, 29630948706756, 131136723532257, 580901892464599, 2575423975663301

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1) )); // G. C. Greubel, Apr 29 2019

CoefficientList[Series[1/Sqrt[x^6+2x^5+x^4-2x^3-2x^2-4x+1], {x, 0, 25}], x] (* Michael De Vlieger, Oct 08 2016 *)

/* same as in A092566 but use */
steps=[[0,1], [1,0], [2,2], [3,3]];
/* Joerg Arndt, Jun 30 2011 */

my(x='x+O('x^30)); Vec(1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1)) \\ G. C. Greubel, Apr 29 2019

(1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

G.f.: 1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1). - Mark van Hoeij, Apr 17 2013

D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+1)*a(n-2) +(-2*n+3)*a(n-3) +(n-2)*a(n-4) +(2*n-5)*a(n-5) +(n-3)*a(n-6)=0. - R. J. Mathar, Oct 08 2016

A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).

Original entry on oeis.org

1, 2, 6, 30, 154, 768, 3906, 20232, 105750, 556328, 2943432, 15646932, 83500126, 447057380, 2400249624, 12918250836, 69674241654, 376489511460, 2037768450480, 11045915485740, 59955446568276, 325821729044784, 1772588671356204, 9653187691115640, 52617711157401186, 287051310425050668

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

Diagonal of rational function 1/(1 - (x + y + x^3 + y^3)). - Gheorghe Coserea, Aug 06 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1344 (first 304 terms from Gheorghe Coserea)

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.

REL := 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1);
ogf := sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3);
series(eval(ogf, s=RootOf(REL,s)),x=0,30);  # Mark van Hoeij, Apr 17 2013
# second Maple program:
b:= proc(x, y) option remember; `if`(y=0, 1, add((p->
      `if`(p[1]<0, 0, b(p[1], p[2])))(sort([x, y]-h)),
        h=[[1, 0], [0, 1], [3, 0], [0, 3]]))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 28 2018

a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k-1] + a[n, k-3] + a[n-1, k] + a[n-3, k]; a[, ] = 0;
a[n_] := a[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Oct 06 2019 *)

/* same as in A092566 but use */
steps=[[1,0], [3,0], [0,1], [0,3]];
/* Joerg Arndt, Jun 30 2011 */

seq(N) = {
  my(x='x + O('x^N), d=16*x^6 + 16*x^5 + 16*x^4 - 8*x^3 - 4*x^2 + 1,
     s=serreverse((1 - 2*x^2 + 2*x^3 - sqrt(d))/(6*x^2)));
  Vec(sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3));
};
seq(26) \\ Gheorghe Coserea, Aug 06 2018

G.f.: sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3)  where s is a function satisfying 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1)=0. - Mark van Hoeij, Apr 17 2013
From Gheorghe Coserea, Aug 06 2018: (Start)
G.f. y=A(x) satisfies:
0 = (4*x^3 + 8*x^2 + 4*x - 1)^4*(108*x^3 - 108*x^2 + 36*x - 31)^2*y^8 + 4*(4*x^3 + 8*x^2 + 4*x - 1)^3*(36*x^3 + 36*x^2 - 4*x - 13)*(108*x^3 - 108*x^2 + 36*x - 31)*y^6 + 2*(4*x^3 + 8*x^2 + 4*x - 1)^2*(2160*x^6 + 4320*x^5 + 1872*x^4 - 1784*x^3 - 1576*x^2 + 472*x + 431)*y^4 + 4*(4*x^3 + 8*x^2 + 4*x - 1)*(112*x^6 + 448*x^5 + 688*x^4 + 456*x^3 + 96*x^2 + 40*x + 55)*y^2 + (4*x^3 + 12*x^2 + 12*x + 3)^2.
0 = (4*x^3 + 8*x^2 + 4*x - 1)*(108*x^3 - 108*x^2 + 36*x - 31)*(270*x^4 + 180*x^3 + 144*x^2 - 225*x - 59)*y''' + (1283040*x^9 + 1924560*x^8 + 1080864*x^7 - 1425816*x^6 - 2135376*x^5 + 33048*x^4 + 702468*x^3 + 134520*x^2 + 43892*x + 30575)*y'' + 30*(111780*x^8 + 149040*x^7 + 120960*x^6 - 122094*x^5 - 172206*x^4 - 6012*x^3 + 36615*x^2 + 10298*x - 1541)*y' + 60*(29160*x^7 + 34020*x^6 + 36288*x^5 - 43092*x^4 - 45882*x^3 - 6462*x^2 + 1890*x + 913)*y.
(End)

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 4, 6, 1, 3, 7, 13, 22, 1, 4, 11, 24, 46, 80, 1, 5, 16, 40, 86, 166, 296, 1, 6, 22, 62, 148, 314, 610, 1106, 1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 1, 8, 37, 128, 367, 920, 2083, 4352, 8518, 15792, 1, 9, 46, 174, 541, 1461, 3544, 7896

Offset: 0

Peter Luschny, Nov 14 2012

Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.
Row sums: [A026641 ] [1,  1,  4,  13,  46,  166,   610]
--
T(j+2, 2) [A000124*] [1*, 2 , 4,   7,  11,  16,     22]
T(j+3, 3) [A003600*] [1*, 2*, 6,  13,  24,  40,     62]
--
T(j  , j) [A072547 ] [1,  0,  2,   6,  22,  80,    296]
T(j+1, j) [A026641 ] [1,  1,  4,  13,  46,  166,   610]
T(j+2, j) [A014300 ] [1,  2,  7,  24,  86,  314,  1163]
T(j+3, j) [A014301*] [1,  3, 11,  40, 148,  553,  2083]
T(j+4, j) [A172025 ] [1,  4, 16,  62, 239,  920,  3544]
T(j+5, j) [A172061 ] [1,  5, 22,  91, 367, 1461,  5776]
T(j+6, j) [A172062 ] [1,  6, 29, 128, 541, 2232,  9076]
T(j+7, j) [A172063 ] [1,  7, 37, 174, 771, 3300, 13820]
--
T(2j  ,j) [Central ] [1,  1,  7,  40, 239, 1461,  9076]
T(2j+1,j) [A183160 ] [1,  2, 11,  62, 367, 2232, 13820]
T(2j+2,j) [        ] [1,  3, 16,  91, 541, 3300, 20476]
T(2j+3,j) [A199033*] [1,  4, 22, 128, 771, 4744, 29618]

			Triangle begins as:
[n]|k->
[0] 1
[1] 1, 0
[2] 1, 1,  2
[3] 1, 2,  4,  6
[4] 1, 3,  7, 13,  22
[5] 1, 4, 11, 24,  46,  80
[6] 1, 5, 16, 40,  86, 166, 296
[7] 1, 6, 22, 62, 148, 314, 610, 1106.

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

A201635 := proc(n,k) option remember; local j;
if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
else add(A201635(n-1,j), j=0..k) fi end:
for n from 0 to 7 do seq(A(n,k), k=0..n) od;

T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2019 *)

{T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 27 2019

@CachedFunction
def A201635(n, k):
    if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
    return add(A201635(n-1, j) for j in (0..k))
for n in (0..7) : [A201635(n, k) for k in (0..n)]

A226952 Triangle of coefficients of Faber polynomials for (3x - sqrt(x^2 - 4x))/2.

Original entry on oeis.org

0, -1, 1, -1, -2, 1, -4, 0, -3, 1, -13, -4, 2, -4, 1, -46, -10, -5, 5, -5, 1, -166, -36, -6, -8, 9, -6, 1, -610, -126, -28, 0, -14, 14, -7, 1, -2269, -456, -92, -24, 10, -24, 20, -8, 1, -8518, -1674, -333, -63, -27, 27, -39, 27, -9, 1

Offset: 0

Dmitry Kruchinin, Jun 24 2013

			Triangle begins as:
    0;
   -1,   1;
   -1,  -2,  1;
   -4,   0, -3,  1;
  -13,  -4,  2, -4,  1;
  -46, -10, -5,  5, -5,  1;

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

[[n eq 0 and k eq 0 select 0 else k eq n select 1 else n*(&+[ (-1)^j*j*Binomial(j+k,k)*Binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)): j in [1..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 29 2019

T[n_,k_]:= If[n==k==0, 0, If[k==n, 1, n*Sum[(-1)^j*j*Binomial[j+k, k]* Binomial[2*n-2*k-j-1, n-k-1]/((j+k)*(n-k)), {j, 1, n-k}]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 29 2019 *)

T(n,k):=if n=0 and k=0 then 0 else if n=k then 1 else n*sum(binomial(i+k,k)*(i)*binomial(2*(n-k)-i-1,n-k-1)*(-1)^(i)/((i+k)*(n-k)),i,1,n-k);

{T(n,k) = if(n==0 && k==0, 0, if(k==n, 1, n*sum(j=1,n-k, (-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)))))}; \\ G. C. Greubel, Apr 29 2019

def T(n, k):
  if (k==n==0): return 0
  elif (k==n): return 1
  else: return n*sum((-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)) for j in (1..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 29 2019

G.f.: log(1 + (1 - sqrt(1-4*t))/2 - t*x) = Sum_{n>0} Sum_{k=0..n} T(n,k) * x^k * t^n/n.

T(n,k) = n*Sum_{j=1..n-k} binomial(j+k,k)*(j)*binomial(2*(n-k)-j-1, n-k-1)*(-1)^j/((j+k)*(n-k)), k

(-1)^(n+1) * Sum_{k=0..n} T(n,k) = 2*A181933(n).

T(n,0) = -A026641(n-1), n>0.

A253571 Total number of even outdegree nodes among all labeled rooted trees on n nodes.

Original entry on oeis.org

1, 2, 15, 144, 1765, 26400, 466459, 9508352, 219651849, 5671088640, 161833149511, 5058050224128, 171837337744813, 6304955850432512, 248477268083174355, 10467916801317273600, 469451601966727952401, 22329535184262444220416, 1122809130124800181976575

Offset: 1

Marko Riedel, Jan 03 2015

nonn

			When n=3 there are two types of trees: rooted paths on three nodes which have one even degree node (the bottom one with zero children), giving 6*1, and trees consisting of a node with two children, of which there are 3, and they have 3 even degree nodes, giving 3*3 for a total of 6*1 + 3*3 = 15.

Alois P. Heinz, Table of n, a(n) for n = 1..380 (first 100 terms from Marko Riedel)
Marko Riedel, Even outdegree nodes among all labeled trees on n nodes

Cf. A000169, A026641, A000081.

a:= n-> n!*coeff(series((T->(T^2+x^2)/
    (2*T*(1-T)))(-LambertW(-x)), x, n+2), x, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 03 2015

E.g.f.: (T^2+z^2)/(2*T*(1-T)) where T is the labeled tree function defined by T = z exp T.

A368489 a(n) = Sum_{k=0..n} n^k * binomial(k+n,k).

Original entry on oeis.org

1, 3, 31, 643, 20421, 873806, 46994011, 3042431715, 230249448841, 19940350062394, 1944516598602711, 210829412453667998, 25156743053019602701, 3275876521195372322892, 462262670054775645538099, 70264375447526610838701091

Offset: 0

Seiichi Manyama, Dec 27 2023

Cf. A001700, A026641, A368488.

a(n) = sum(k=0, n, n^k*binomial(k+n, k));

a(n) = [x^n] 1/((1-x) * (1-n*x)^(n+1)).

a(n) ~ 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Dec 27 2023

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A382848 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(n+k,k).

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A188289 Binomial sum related to rooted trees.

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A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).

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A192371 Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).

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A192417 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,2), (3,3).

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A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).

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A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

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A226952 Triangle of coefficients of Faber polynomials for (3x - sqrt(x^2 - 4x))/2.