A005717 -id:A005717 - cp's OEIS frontend

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

0, 0, 1, 1, 5, 11, 34, 92, 265, 751, 2156, 6194, 17874, 51702, 149941, 435749, 1268761, 3700391, 10808548, 31613474, 92577784, 271407896, 796484503, 2339561795, 6877992334, 20236257626, 59581937299, 175546527727, 517538571125, 1526679067331, 4505996000730

Offset: 0

Peter Luschny, Aug 30 2011

nonn

The inverse binomial transform of a(n) is A194590(n).

Peter Luschny, The lost Catalan numbers.

Cf. A005043, A189912, A194588, A100071, A005717.

# First method, describes the derivation:
A056040 := n -> n!/iquo(n,2)!^2:
A057977 := n -> A056040(n)/(iquo(n,2)+1);
A001006 := n -> add(binomial(n,k)*A057977(k)*irem(k+1,2),k=0..n):
A005043 := n -> `if`(n=0,1,A001006(n-1)-A005043(n-1)):
A189912 := n -> add(binomial(n,k)*A057977(k),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):
A194589 := n -> A194588(n)-A005043(n):
# Second method, more efficient:
A100071 := n -> A056040(n)*(n/2)^(n-1 mod 2):
A194589 := proc(n) local k;
(n mod 2)+(1/2)*add((-1)^k*binomial(n,k)*A100071(k+1),k=1..n) end:
# Alternatively:
a := n -> `if`(n<3,iquo(n,2),hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4)): seq(simplify(a(n)), n=0..30); # Peter Luschny, Mar 07 2017

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Mod[n, 2] + (1/2)*Sum[(-1)^k*Binomial[n, k]*2^-Mod[k, 2]*(k+1)^Mod[k, 2]*sf[k+1], {k, 1, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 30 2013, from 2nd method *)
Table[If[n < 3, Quotient[n, 2], HypergeometricPFQ[{1 - n/2, -n, 3/2 - n/2}, {1, 2-n}, 4]], {n,0,30}] (* Peter Luschny, Mar 07 2017 *)

a(n):=sum(binomial(n+2,k)*binomial(n-k,k),k,0,(n)/2); /* Vladimir Kruchinin, Sep 28 2015 */

a(n) = sum(k=0, n/2, binomial(n+2,k)*binomial(n-k,k));
vector(30, n, a(n-3)) \\ Altug Alkan, Sep 28 2015

a(n) = sum_{k=0..n} C(n,k)*A194590(k).
a(n) = (n mod 2)+(1/2)*sum_{k=1..n} (-1)^k*C(n,k)*(k+1)$*((k+1)/2)^(k mod 2). Here n$ denotes the swinging factorial A056040(n).
a(n) = PSUMSIGN([0,0,1,2,6,16,45,..] = PSUMSIGN([0,0,A005717]) where PSUMSIGN is from Sloane's "Transformations of integer sequences". - Peter Luschny, Jan 17 2012
A(x) = B'(x)*(1/x^2-1/(B(x)*x)), where B(x)/x is g.f. of A005043. - Vladimir Kruchinin, Sep 28 2015
a(n) = Sum_{k=0..n/2} C(n+2,k)*C(n-k,k). - Vladimir Kruchinin, Sep 28 2015
a(n) = hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4) for n>=3. - Peter Luschny, Mar 07 2017
a(n) ~ 3^(n + 1/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 17 2024

A219670 Number of n-step paths on cubic lattice from (0,0,0) to (1,0,0) with moves in any direction on {-1,0,1}^3 and zero moves allowed.

Original entry on oeis.org

0, 1, 18, 294, 5776, 117045, 2505006, 55138293, 1245056184, 28643604147, 669304345150, 15838583011812, 378828554265096, 9143273873757283, 222407411228180010, 5446827816890184990, 134191612737844924608, 3323506599627088488579, 82700482246125321972582

Offset: 0

Jon Perry, Nov 24 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A219671, A219986.

Cf. A002426, A005717.

b=[[1,1,1],[1,1,0],[1,1,-1],[1,0,1],[1,0,0],[1,0,-1],[1,-1,1],[1,-1,0],[1,-1,-1],
[0,1,1],[0,1,0],[0,1,-1],[0,0,1],[0,0,0],[0,0,-1],[0,-1,1],[0,-1,0],[0,-1,-1],
[-1,1,1],[-1,1,0],[-1,1,-1],[-1,0,1],[-1,0,0],[-1,0,-1],[-1,-1,1],[-1,-1,0],[-1,-1,-1]];
function inc(arr,m) {
al=arr.length-1;
full=true;
for (ac=0;ac<=al;ac++) if (arr[ac]!=m) {full=false;break;}
if (full==true) return false;
while (arr[al]==m && al>0) {arr[al]=0;al--;}
arr[al]++;
return true;
}
for (k=0;k<6;k++) {
c=0;
a=new Array();
for (i=0;i

a:= proc(n) a(n):= `if`(n<6, [0, 1, 18, 294, 5776, 117045][n+1],
     (n*(n-1)*(453658*n^4-2664929*n^3+6608535*n^2-8353208*n+3876664)
      *a(n-1) +3*(n-1)*(286527*n^5+2962040*n^4-19850405*n^3+25517846
      *n^2+20905560*n-41336424) *a(n-2) -18*(n-2)*(2*n-5)*(1294945*n^4
      -12949450*n^3+54428897*n^2-110276360*n+88672932) *a(n-3) -81*(n-3)
      *(286527*n^5-10125215*n^4+111022145*n^3-530226521*n^2+1163720520*n
      -966508776) *a(n-4) +729*(453658*n^4-6408231*n^3+34683300*n^2
      -84691467*n+77744124)*(n-4)^2 *a(n-5) +19683*(-42552+15593*n)
      *(n-4)^2 *(n-5)^3 *a(n-6))/ (n^2*(n+1)*(n-1)^2*(15593*n-35413)))
    end:
seq (a(n), n=0..30);  # Alois P. Heinz, Nov 28 2012
A005717 := n -> simplify(GegenbauerC(n-1,-n,-1/2));
A002426 := n -> simplify(GegenbauerC(n,-n,-1/2));
seq( A002426(n)^2 * A005717(n), n=0..30 );  # Mark van Hoeij, Nov 13 2022

a(n) ~ 3^(3*n+3/2) / (4*Pi*n)^(3/2). - Vaclav Kotesovec, Sep 07 2014
Recurrence (of order 4): (n-1)^2*n^2*(n+1)*(2*n-5)*(7*n^4 - 56*n^3 + 166*n^2 - 216*n + 105)*a(n) = (n-1)*n*(2*n-5)*(2*n-1)*(70*n^6 - 630*n^5 + 2206*n^4 - 3759*n^3 + 3181*n^2 - 1188*n + 144)*a(n-1) + 3*(n-1)*(2*n-3)*(490*n^8 - 5880*n^7 + 30030*n^6 - 85050*n^5 + 145359*n^4 - 152064*n^3 + 93599*n^2 - 30264*n + 3852)*a(n-2) - 27*(n-2)^2*(2*n-5)*(2*n-1)*(70*n^6 - 630*n^5 + 2206*n^4 - 3813*n^3 + 3424*n^2 - 1563*n + 342)*a(n-3) - 729*(n-3)^3*(n-2)^2*(2*n-1)*(7*n^4 - 28*n^3 + 40*n^2 - 24*n + 6)*a(n-4). - Vaclav Kotesovec, Sep 07 2014
a(n) = A002426(n)^2 * A005717(n). - Mark van Hoeij, Nov 13 2022

More terms from Alois P. Heinz, Nov 28 2012

A371408 Number of Dyck paths of semilength n having exactly three (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 20, 80, 315, 1176, 4284, 15240, 53295, 183700, 625768, 2110472, 7057505, 23427600, 77271120, 253426752, 827009523, 2686728060, 8693388060, 28026897360, 90058925649, 288516259416, 921755412900, 2937377079000, 9338728806225, 29626186593276

Offset: 0

Alois P. Heinz, Mar 22 2024

nonn

			a(4) = 1: UDUDUDUD.
a(5) = 4: UDUDUDUUDD, UDUDUUDUDD, UDUUDUDUDD, UUDUDUDUDD.

Alois P. Heinz, Table of n, a(n) for n = 0..2090
Wikipedia, Counting lattice paths

Column k=3 of A091869.

Cf. A000108, A001006, A005717, A102839, A121262.

a:= n-> `if`(n<4, 0, binomial(n-1, 3)*add(binomial(n-3, j)*
         binomial(n-3-j, j-1), j=0..ceil((n-3)/2))/(n-3)):
seq(a(n), n=0..29);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0$4, 1][n+1],
     (n-1)*((2*n-7)*a(n-1)+3*(n-2)*a(n-2))/((n-2)*(n-4)))
    end:
seq(a(n), n=0..29);

a(n) mod 2 = A121262(n) for n >= 1.

A375253 Expansion of (1 - 2x + 2x^2)/(1 - 2x - 3x^2)^(7/2).

Original entry on oeis.org

1, 5, 30, 140, 630, 2646, 10710, 41910, 159885, 597025, 2190188, 7914270, 28230020, 99567300, 347720040, 1203777072, 4135047615, 14105322315, 47813634330, 161154659820, 540353553894, 1803226621350, 5991410183850, 19827295283250, 65371101643575

Offset: 0

Seiichi Manyama, Aug 07 2024

nonn

Column k=4 of A091869 (with a different offset).
Cf. A374506, A375248.
Cf. A005717, A373651.

a[n_]:=(1+n)(2+n)(3+n)(4+n)Hypergeometric2F1[(1-n)/2,-n/2,2,4]/24; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

my(N=30, x='x+O('x^N)); Vec((1-2*x+2*x^2)/(1-2*x-3*x^2)^(7/2))

a(n) = (binomial(n+4,3)/4) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = (binomial(n+4,3)/4) * A005717(n+1).
a(n) = ((n+4)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+3)*a(n-2)).
a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([(1-n)/2, -n/2], [2], 4)/24. - Stefano Spezia, Aug 07 2024

A375259 Expansion of (1 - 3x + 6x^2 - 4x^3)/(1 - 2x - 3*x^2)^(9/2).

Original entry on oeis.org

1, 6, 42, 224, 1134, 5292, 23562, 100584, 415701, 1671670, 6570564, 25325664, 95982068, 358442280, 1321336152, 4815108288, 17367199983, 62063418186, 219942717918, 773542367136, 2701767769470, 9376778431020, 32353614992790, 111032853586200, 379152389532735

Offset: 0

Seiichi Manyama, Aug 08 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=5 of A091869 (with a different offset).

Cf. A005717, A375260.

my(N=30, x='x+O('x^N)); Vec((1-3*x+6*x^2-4*x^3)/(1-2*x-3*x^2)^(9/2))

a(n) = (binomial(n+5,4)/5) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = (binomial(n+5,4)/5) * A005717(n+1).
a(n) = ((n+5)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+4)*a(n-2)).

A098470 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.

Original entry on oeis.org

1, 6, 28, 112, 414, 1452, 4917, 16236, 52624, 168168, 531531, 1665456, 5182008, 16031952, 49366674, 151419816, 462919401, 1411306358, 4292487562, 13029127584, 39478598170, 119439969220, 360881425710, 1089126806040

Offset: 5

Eric W. Weisstein, Sep 09 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 5..1005
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Cf. A002426, A005717, A014531, A014532, A014533.

# Assuming offset 0:
a := n -> simplify(GegenbauerC(n, -n-5, -1/2)):
seq(a(n), n=0..25); # Peter Luschny, May 09 2016

Table[GegenbauerC[n, -n - 5, -1/2], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)

x='x + O('x^50); Vec(32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5)) \\ G. C. Greubel, Feb 28 2017

(n^2-25)*a(n) = n*(2*n-1)*a(n-1) + 3*n*(n-1)*a(n-2). - Vladeta Jovovic, Sep 18 2004
G.f.: 32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5). - Vladeta Jovovic, Sep 18 2004
a(n) = A111808(n,n-5). - Reinhard Zumkeller, Aug 17 2005
Assuming offset 0: a(n) = GegenbauerC(n,-n-5,-1/2) and a(n) = binomial(10+2*n,n)* hypergeom([-n, -n-10], [-9/2-n], 1/4). - Peter Luschny, May 09 2016
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 09 2021

A104632 1/n times A104631(n), the coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

Original entry on oeis.org

1, 2, 6, 20, 73, 281, 1125, 4635, 19525, 83710, 364070, 1602327, 7123041, 31937010, 144255802, 655804649, 2998354717, 13777825186, 63596593430, 294743653360, 1371017707245, 6398580086645, 29952930770185, 140604572777250, 661708404611603, 3121439743413256, 14756658303857332

Offset: 1

T. D. Noe, Mar 17 2005

This sequence may be viewed as a higher-order form of the Motzkin numbers, A001006, which are 1/n times the coefficient of x^(n+1) in the expansion of (1+x+x^2)^n. According to Superseeker, this sequence is the INVERT transform of A104184, which is related to Motzkin numbers also. See A104631 for additional comments.

Alternatively, this sequence corresponds to the number of positive walks with n steps {-2,-1,0,1,2} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A005717 (coefficient of x^(n+1) in the expansion of (1+x+x^2)^n).

f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1]/n, {n, 30}]
a[ n_] := If[ n < 1, 0, Coefficient[ (1 + x + x^2 + x^3 + x^4)^n, x, 2 n + 1] / n]; (* Michael Somos, Dec 01 2016 *)

a(n):=sum((-1)^i*binomial(n,i)*binomial(3*n-5*i,n-1),i,0,(2*n+1)/5)/n; /* Vladimir Kruchinin, Apr 06 2017 */

a(n) = polcoeff((1+x+x^2+x^3+x^4)^n, 2*n+1)/n \\ Michel Marcus, Sep 24 2016

a(n) = Sum_{i=0..(2*n+1)/5}((-1)^i*binomial(n,i)*binomial(3*n-5*i,n-1))/n. - Vladimir Kruchinin, Apr 06 2017

Conjecture: 2*n*(2*n+1)*(n-1)*a(n) -(n-1)*(19*n^2-19*n+2)*a(n-1) -5*(n-2)*(2*n^2-3*n-1)*a(n-2) +25*n*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 23 2017

A114583 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)).

Original entry on oeis.org

1, 1, 2, 3, 1, 7, 2, 15, 6, 36, 14, 1, 85, 39, 3, 209, 102, 12, 517, 280, 37, 1, 1303, 758, 123, 4, 3312, 2085, 381, 20, 8510, 5730, 1194, 76, 1, 22029, 15849, 3657, 295, 5, 57447, 43914, 11187, 1056, 30, 150709, 122090, 33903, 3734, 135, 1, 397569, 340104

Offset: 0

Emeric Deutsch, Dec 09 2005

Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 1 yields A114584. Sum(k*T(n,k),k=0..floor(n/3))=A005717(n-2).

			T(5,1)=6 because we have HH(UHD), UD(UHD), (UHD)HH, (UHD)UD, H(UHD)H and U(UHD)D, where U=(1,1),H=(1,0),D=(1,-1) (the UHD's are shown between parentheses).
Triangle begins:
   1;
   1;
   2;
   3,  1;
   7,  2;
  15,  6;
  36, 14, 1;
  ...

Alois P. Heinz, Rows n = 0..300, flattened
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

Cf. A001006, A114584, A115717.

G:=(1-z-t*z^3+z^3-sqrt((1-3*z+z^3-t*z^3)*(1+z+z^3-t*z^3)))/2/z^2: Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 17 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y, `if`(t=1, 2, 0))+b(x-1, y-1, 0)*
     `if`(t=2, z, 1)+b(x-1, y+1, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 01 2019

CoefficientList[#, t]& /@ CoefficientList[(1 - z - t z^3 + z^3 - Sqrt[(1 - 3z + z^3 - t z^3)(1 + z + z^3 - t z^3)])/2/z^2 + O[z]^17, z] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

G.f.=G=G(t, z) satisfies G=1+zG+z^2*G(tz-z+G).

A128015 Binomial coefficients C(2n+1,n) repeated.

Original entry on oeis.org

1, 1, 3, 3, 10, 10, 35, 35, 126, 126, 462, 462, 1716, 1716, 6435, 6435, 24310, 24310, 92378, 92378, 352716, 352716, 1352078, 1352078, 5200300, 5200300, 20058300, 20058300, 77558760, 77558760, 300540195, 300540195

Offset: 0

Paul Barry, Feb 11 2007

Hankel transform is A128017. Binomial transform is A005717(n+1).

Cf. A000108, A001700, A005717, A128017.

With[{c=Table[Binomial[2n+1,n],{n,0,20}]},Riffle[c,c]] (* Harvey P. Dale, May 02 2012 *)

G.f.: (1+x)*c(x^2)/sqrt(1-4x^2), c(x) the g.f. of A000108.
E.g.f.: exp(-x)*dif(exp(x)*Bessel_I(1,2x),x).
a(n) = C(n+1, n/2)*(1+(-1)^n)/2 + C(n, (n-1)/2)*(1-(-1)^n)/2; as moment sequence a(n) = (1/(2*Pi))*Integral_{x=-2..2} x^n*x*(1+x)/sqrt(4-x^2).
D-finite with recurrence: -(n+2)*(3*n-1)*a(n) - 4*a(n-1) + 4*n*(3*n+2)*a(n-2) = 0. - R. J. Mathar, Jun 17 2016
a(n) = A001700(floor(n/2)). - Georg Fischer, Nov 28 2022

A194586 Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)A056040(k)(k mod 2)*q^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 3, 0, 6, 0, 4, 0, 24, 0, 0, 5, 0, 60, 0, 30, 0, 6, 0, 120, 0, 180, 0, 0, 7, 0, 210, 0, 630, 0, 140, 0, 8, 0, 336, 0, 1680, 0, 1120, 0, 0, 9, 0, 504, 0, 3780, 0, 5040, 0, 630, 0, 10, 0, 720, 0, 7560, 0, 16800, 0, 6300, 0, 0, 11, 0, 990, 0, 13860, 0, 46200, 0, 34650, 0, 2772, 0, 12

Offset: 0

Peter Luschny, Aug 29 2011

Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the complementary Motzkin numbers A005717. (See A089627 for the Motzkin numbers and A163649 for the extended Motzkin numbers.)

			               0
              0, 1
            0, 2, 0
           0, 3, 0, 6
         0, 4, 0, 24, 0
       0, 5, 0, 60, 0, 30
    0, 6, 0, 120, 0, 180, 0
  0, 7, 0, 210, 0, 630, 0, 140
                0
                q
               2 q
            3 q + 6 q^3
           4 q + 24 q^3
       5 q + 60 q^3  + 30 q^5
      6 q + 120 q^3  + 180 q^5
  7 q + 210 q^3  + 630 q^5  + 140 q^7

Peter Luschny, The lost Catalan numbers.

Row sums are A109188. Cf. A056040, A005717, A163649, A089627.

A194586 := proc(n,k) local j, swing; swing := n -> n!/iquo(n,2)!^2:
add(binomial(n,j)*swing(j)*q^j*(j mod 2),j=0..n); coeff(%,q,k) end:
seq(print(seq(A194586(n,k),k=0..n)),n=0..8);

sf[n_] := n!/Quotient[n, 2]!^2;
row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;
Table[row[n], {n, 0, 12}] (* Jean-François Alcover, Jun 26 2019 *)

egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).

cp's OEIS Frontend

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

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A219670 Number of n-step paths on cubic lattice from (0,0,0) to (1,0,0) with moves in any direction on {-1,0,1}^3 and zero moves allowed.

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A371408 Number of Dyck paths of semilength n having exactly three (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

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Maple

Formula

A375253 Expansion of (1 - 2*x + 2*x^2)/(1 - 2*x - 3*x^2)^(7/2).

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Mathematica

PARI

Formula

A375259 Expansion of (1 - 3*x + 6*x^2 - 4*x^3)/(1 - 2*x - 3*x^2)^(9/2).

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PARI

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A098470 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.

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Maple

Mathematica

PARI

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A104632 1/n times A104631(n), the coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

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Mathematica

Maxima

PARI

Formula

A114583 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)).

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A375253 Expansion of (1 - 2x + 2x^2)/(1 - 2x - 3x^2)^(7/2).

A375259 Expansion of (1 - 3x + 6x^2 - 4x^3)/(1 - 2x - 3*x^2)^(9/2).

A194586 Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)A056040(k)(k mod 2)*q^k.