A214292 -id:A214292 - cp's OEIS frontend

A064059 Seventh column of Catalan triangle A009766.

132, 429, 1001, 2002, 3640, 6188, 9996, 15504, 23256, 33915, 48279, 67298, 92092, 123970, 164450, 215280, 278460, 356265, 451269, 566370, 704816, 870232, 1066648, 1298528, 1570800, 1888887, 2258739, 2686866, 3180372, 3746990, 4395118, 5133856, 5973044

Offset: 0

Wolfdieter Lang, Sep 13 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000096, A005586, A005587, A005557 (third to sixth column).

A064059:= func< n | (n+1)*Binomial(n+12,5)/6 >;
[A064059(n): n in [0..40]]; // G. C. Greubel, Sep 27 2024

[seq(binomial(n+1,6)-2*binomial(n,5),n=12..55)]; # Zerinvary Lajos, Jul 19 2006

CoefficientList[Series[(42 z^5-252 z^4+616 z^3-770 z^2+495 z-132)/(z-1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{132,429,1001,2002,3640,6188,9996},40] (* Harvey P. Dale, Jan 08 2025 *)

def A064059(n): return (n+1)*binomial(n+12,5)//6
[A064059(n) for n in range(41)] # G. C. Greubel, Sep 27 2024

G.f.: (132-495*x+770*x^2-616*x^3+252*x^4-42*x^5)/(1-x)^7; numerator polynomial is N(2;5, x) from A062991.
a(n) = A009766(n+6, 6) = (n+1)*binomial(n+12,5)/6.
a(n) = binomial(n+13,6) - 2*binomial(n+12,5). - Zerinvary Lajos, Jul 19 2006
a(n) = A214292(n+11,5). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 25961/2134440.
Sum_{n>=0} (-1)^n/a(n) = 4160*log(2)/77 - 79917773/2134440. (End)

A000590 a(n) = 13*binomial(2n,n-6)/(n+7).

Original entry on oeis.org

1, 13, 104, 663, 3705, 19019, 92092, 427570, 1924065, 8454225, 36463440, 154969620, 650872404, 2707475148, 11173706960, 45812198536, 186803188858, 758201178306, 3065415516592, 12352414499425, 49634247352235, 198954083924505, 795816335698020, 3177498557750790

Offset: 6

N. J. A. Sloane

nonn

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=6. - Herbert Kociemba, May 24 2004

Number of standard tableaux of shape (n+6,n-6). - Emeric Deutsch, May 30 2004

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 6..200
Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), Article 00.1.6.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

Cf. A000108, A001622, A214292.

a[n_] := 13*Binomial[2*n, n-6]/(n+7); Array[a, 24, 6] (* Amiram Eldar, Sep 26 2022 *)

a(n) = 13*binomial(2*n,n-6)/(n+7); \\ Michel Marcus, Oct 16 2017

G.f.: x^6*C(x)^13, where C(x)=[1-sqrt(1-4x)]/(2x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=12, a(n-6)=(-1)^(n-12)*coeff(charpoly(A,x),x^12). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n-1,n-7) for n > 6. - Reinhard Zumkeller, Jul 12 2012
-(n+7)*(n-6)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jun 20 2013
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=6} 1/a(n) = 16777/5460 - 128*Pi/(117*sqrt(3)).
Sum_{n>=6} (-1)^n/a(n) = 787536*log(phi)/(325*sqrt(5)) - 14210999/27300, where phi is the golden ratio (A001622). (End)

A005557 Number of walks on square lattice.

Original entry on oeis.org

42, 132, 297, 572, 1001, 1638, 2548, 3808, 5508, 7752, 10659, 14364, 19019, 24794, 31878, 40480, 50830, 63180, 77805, 95004, 115101, 138446, 165416, 196416, 231880, 272272, 318087, 369852, 428127, 493506, 566618, 648128, 738738, 839188, 950257, 1072764

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Sixth diagonal of Catalan triangle A033184.
Sixth column of Catalan triangle A009766.
Cf. A062991, A214292.

List([0..30],n->(n+1)*Binomial(n+10,4)/5); # Muniru A Asiru, Apr 10 2018

[(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

[seq(binomial(n,5)-binomial(n,3),n=9..55)]; # Zerinvary Lajos, Jul 19 2006
A005557:=(42-120*z+135*z**2-70*z**3+14*z**4)#(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(14 z^4 - 70 z^3 + 135 z^2 - 120 z + 42)/(z - 1)^6, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{42,132,297,572,1001,1638},40] (* Harvey P. Dale, Feb 22 2024 *)

a(n)=(n+1)*binomial(n+10,4)/5 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = A009766(n+5, 5) = (n+1)*binomial(n+10, 4)/5.
G.f.: (42 - 120*x + 135*x^2 - 70*x^3 + 14*x^4)/(1-x)^6; numerator polynomial is N(2;4, x) from A062991.
a(n) = binomial(n+9,5) - binomial(n+9,3). - Zerinvary Lajos, Jul 19 2006
a(n) = A214292(n+9, 4). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 2509/63504.
Sum_{n>=0} (-1)^n/a(n) = 951395/63504 - 1360*log(2)/63. (End)

More terms and formula from Wolfdieter Lang, Sep 04 2001

A064061 Eighth column of Catalan triangle A009766.

Original entry on oeis.org

429, 1430, 3432, 7072, 13260, 23256, 38760, 62016, 95931, 144210, 211508, 303600, 427570, 592020, 807300, 1085760, 1442025, 1893294, 2459664, 3164480, 4034712, 5101360, 6399888, 7970688, 9859575, 12118314, 14805180, 17985552, 21732542, 26127660, 31261516

Offset: 0

Wolfdieter Lang, Sep 13 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A009766, A064059 (seventh column), A062991, A214292.

A064061:= func< n | (n+1)*Binomial(n+14, 6)/7 >;
[A064061(n): n in [0..40]]; // G. C. Greubel, Sep 28 2024

[seq(binomial(n,7)-binomial(n,5),n=13..37)]; # Zerinvary Lajos, Nov 25 2006

CoefficientList[Series[(132*z^6 - 924*z^5 + 2730*z^4 - 4368*z^3 + 4004*z^2 - 2002*z + 429)/(z - 1)^8, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
Table[Binomial[n,7]-Binomial[n,5],{n,13,50}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{429,1430,3432,7072,13260,23256,38760,62016},40] (* Harvey P. Dale, Sep 03 2015 *)

def A064061(n): return (n+1)*binomial(n+14,6)//7
[A064061(n) for n in range(41)] # G. C. Greubel, Sep 28 2024

a(n) = A009766(n+7, 7) = (n+1)*binomial(n+14, 6)/7.
G.f.: (429-2002*x+4004*x^2-4368*x^3+2730* x^4-924*x^5+132*x^6)/(1-x)^8; numerator polynomial is N(2;6, x) from A062991.
a(n) = C(n+13,7) - C(n+13,5). - Zerinvary Lajos, Nov 25 2006
a(n) = A214292(n+13,6). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 323171/88339680.
Sum_{n>=0} (-1)^n/a(n) = 7929257917/88339680 - 55552*log(2)/429. (End)

A129936 a(n) = (n-2)(n+3)(n+2)/6.

Original entry on oeis.org

-2, -2, 0, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733, 8398, 9100, 9840, 10619, 11438, 12298, 13200, 14145, 15134, 16168, 17248

Offset: 0

Roger L. Bagula, Jun 09 2007

Essentially the same as A005586.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A002378, A005586, A023444, A034856, A214292.

seq(sum(i*(k-i+1), i=1..k+2), k=0..99); # Wesley Ivan Hurt, Sep 21 2013

f[n_] = Binomial[n + 3, 3] - (n + 3)*(n + 2)/2; Table[f[n], {n, 0, 30}]
LinearRecurrence[{4,-6,4,-1},{-2,-2,0,5},50] (* Harvey P. Dale, Jul 03 2020 *)

a(n)=(n-2)*(n+3)*(n+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = binomial(n+3,3) - (n + 3)*(n + 2)/2.
a(n) = A214292(n+2,2). - Reinhard Zumkeller, Jul 12 2012
G.f.: (x^3-4*x^2+6*x-2)/(x-1)^4. - Colin Barker, Sep 05 2012
From Wesley Ivan Hurt, Sep 21 2013: (Start)
a(n) = Sum_{i=1..n+2} i*(n-i+1).
a(n+2) = A000292(n+1) + A034856(n), n>0. (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=3} 1/a(n) = 77/200.
Sum_{n>=3} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*(x^3 + 6*x^2 - 12)/6.
a(n) = A023444(n)*A002378(n+2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Wesley Ivan Hurt, Sep 21 2013

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset: 5

Philippe Deléham, Feb 15 2004

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 5..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)

for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, 0, -2, 0, 3, 2, -2, -3, 0, 4, 5, 0, -5, -4, 0, 5, 9, 5, -5, -9, -5, 0, 6, 14, 14, 0, -14, -14, -6, 0, 7, 20, 28, 14, -14, -28, -20, -7, 0, 8, 27, 48, 42, 0, -42, -48, -27, -8, 0, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 0, 10, 44, 110

Offset: 0

Reinhard Zumkeller, Jul 18 2015

sign

A214292 gives first differences per row in Pascal's triangle.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A014473, A163866, A214292.

Cf. A000108, A021499, A047171, A074331.

a259525 n = a259525_list !! n
a259525_list = zipWith (-) (tail pascal) pascal
                           where pascal = concat a007318_tabl

[k eq n select 0 else (n-2*k-1)*Binomial(n,k+1)/(n-k): k in [0..n], n in [0..14]]; // G. C. Greubel, Apr 25 2024

Table[If[k==n, 0, ((n-2*k-1)/(n-k))*Binomial[n,k+1]], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Apr 25 2024 *)

flatten([[binomial(n,k+1) -binomial(n,k) +int(k==n) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Apr 25 2024

From G. C. Greubel, Apr 25 2024: (Start)
If viewed as a triangle then:
T(n, k) = binomial(n, k+1) - binomial(n, k), with T(n, n) = 0.
T(n, n-k) = - T(n, k), for 0 <= k < n.
T(2*n, n) = [n=0] - A000108(n).
Sum_{k=0..n} T(n, k) = 0 (row sums).
Sum_{k=0..floor(n/2)} T(n, k) = A047171(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A021499(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A074331(n-1). (End)

A124087 9th column of Catalan triangle A009766.

Original entry on oeis.org

1430, 4862, 11934, 25194, 48450, 87210, 149226, 245157, 389367, 600875, 904475, 1332045, 1924065, 2731365, 3817125, 5259150, 7152444, 9612108, 12776588, 16811300, 21912660, 28312548, 36283236, 46142811, 58261125, 73066305, 91051857, 112784399, 138912059

Offset: 15

Zerinvary Lajos, Nov 25 2006

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A009766, A214292.

Cf. A064059, A064061, A124088.

[seq(binomial(n,8)-binomial(n,6),n=15..45)];

CoefficientList[Series[(429*z^7 - 3432*z^6 + 11880*z^5 - 23100*z^4 + 27300*z^3 - 19656*z^2 + 8008*z - 1430)/(z - 1)^9, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
Table[Binomial[n,8]-Binomial[n,6],{n,15,60}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1430,4862,11934,25194,48450,87210,149226,245157,389367},30] (* Harvey P. Dale, Apr 15 2017 *)

a(n) = C(n,8)-C(n,6).
a(n) = A214292(n+15,7). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=15} 1/a(n) = 12515/11594583.
Sum_{n>=15} (-1)^(n+1)/a(n) = 1942528*log(2)/6435 - 60651032147/289864575. (End)

A124088 10th column of Catalan triangle A009766.

Original entry on oeis.org

4862, 16796, 41990, 90440, 177650, 326876, 572033, 961400, 1562275, 2466750, 3798795, 5722860, 8454225, 12271350, 17530500, 24682944, 34295052, 47071640, 63882940, 85795600, 114108148, 150391384, 196534195, 254795320, 327861625, 418913482, 531697881

Offset: 17

Zerinvary Lajos, Nov 25 2006

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A009766, A214292.

Cf. A064059, A064061, A124087.

[seq(binomial(n,9)-binomial(n,7),n=17..42)];

CoefficientList[Series[(1430*z^8 - 12870*z^7 + 51051*z^6 - 116688*z^5 + 168300*z^4 - 157080*z^3 + 92820*z^2 - 31824*z + 4862)/(z - 1)^10, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)

a(n) = C(n,9)-C(n,7).
a(n) = A214292(n+17,8). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=17} 1/a(n) = 2074783/6618932320.
Sum_{n>=17} (-1)^(n+1)/a(n) = 2259208566291/4727808800 - 8379648*log(2)/12155. (End)

A128634 Number of parallel permutations of length n.

Original entry on oeis.org

0, 2, 8, 26, 82, 262, 856, 2858, 9722, 33590, 117570, 416022, 1485798, 5348878, 19389688, 70715338, 259289578, 955277398, 3534526378, 13128240838, 48932534038, 182965127278, 686119227298, 2579808294646, 9723892802902, 36734706144302, 139067101832006, 527495903500718

Offset: 1

Ralf Stephan, May 08 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Mansour and S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO], 2006.

Cf. A001453, A068875, A214292.

List([1..30], n-> 2*(Binomial(2*n, n)/(n+1) -1) ); # G. C. Greubel, Dec 02 2019

[2*(Catalan(n)-1): n in [1..40]]; // Vincenzo Librandi, Jul 22 2015

c:=binomial(2*n,n)/(n+1); seq(2*(c(n)-1), n=1..30); # G. C. Greubel, Dec 02 2019

Table[2 (CatalanNumber[n] - 1), {n, 30}] (* Vincenzo Librandi, Jul 22 2015 *)

vector(30, n, 2*(binomial(2*n,n)/(n+1) -1) ) \\ Michel Marcus, Jul 21 2015

[2*(catalan_number(n) -1) for n in (1..30)] # G. C. Greubel, Dec 02 2019

a(n) = -2 + 2 * binomial(2*n,n)/(n+1).
a(n) = -2 + A068875(n+1).
a(n) = 2*A001453(n) for n > 1. - J. M. Bergot, Sep 03 2013
a(n)= Sum_{r=0..n} A214292(n, r)^2. - J. M. Bergot, Sep 04 2013

More terms from Michel Marcus, Jul 21 2015

Offset changed by G. C. Greubel, Dec 02 2019

cp's OEIS Frontend

A064059 Seventh column of Catalan triangle A009766.

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A000590 a(n) = 13*binomial(2n,n-6)/(n+7).

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A005557 Number of walks on square lattice.

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A064061 Eighth column of Catalan triangle A009766.

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A129936 a(n) = (n-2)*(n+3)*(n+2)/6.

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Maple

Mathematica

PARI

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A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

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A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.

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A129936 a(n) = (n-2)(n+3)(n+2)/6.