A052535 -id:A052535 - cp's OEIS frontend

A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 15, 10, 1, 1, 9, 28, 35, 15, 1, 1, 11, 45, 84, 70, 21, 1, 1, 13, 66, 165, 210, 126, 28, 1, 1, 15, 91, 286, 495, 462, 210, 36, 1, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1

Offset: 0

Clark Kimberling

Row sums are odd-indexed Fibonacci numbers.
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - Emeric Deutsch, May 31 2004
Diagonal sums are A052535. - Paul Barry, Jan 21 2005
Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry, Jan 21 2005
Coefficients of Morgan-Voyce polynomial b(n,x); e.g., b(3,x)=x^3+5x^2+6x+1. See A172431 for coefficients of Morgan-Voyce polynomial B(n,x). - Clark Kimberling, Feb 13 2010
T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. - Emanuele Munarini, Apr 07 2011
Roots of signed n-th polynomials are chaotic with respect to the operation (-2, x^2), with cycle lengths A003558(n). Example: starting with a root to x^3 - 5x^2 + 6x - 1 = 0; (2 + 2*cos(2*Pi/N) = 3.24697... = A116415; we obtain the trajectory (3.24697...-> 1.55495...-> 0.198062...; the 3 roots to the polynomial with cycle length 3 matching A003558(3) = 3. The operation (-2, x^2) is the reversal of the well known chaotic operation (x^2 - 2) [Kappraff, Adamson, 2004] starting with seed 2*cos(2*Pi/N). Check: given 2*cos(2*Pi/7) = 1.24697..., we obtain the 3-cycle using (x^2 - 2): (1.24697...-> -0.445041...-> 1.801937...; where the terms in either set are intermediate terms in the other, irrespective of sign. - Gary W. Adamson, Sep 22 2011
A054142 is jointly generated with A172431 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section of A172431. - Clark Kimberling, Mar 09 2012
Subtriangle of the triangle given by (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012
The o.g.f. for row n of the array A(n, k) = binomial(2*n-k,k), k >= 0, n >= 0 is G(n,x) = Sum_{k=0..n} T(n, k)*x^k + (-x)^(2*n+1) * c(-x)^(2*n+1) / sqrt(1-4*(-x)), for n >= 0. Here c(x) is the o.g.f. of A000108 (Catalan). For powers of c(x) see the W. Lang link in A115139. For the alternating sign case replace x by -x. - Wolfdieter Lang, Sep 12 2016
Multiplying the n-th diagonal by A001147(n) generates A001497. - Tom Copeland, Oct 04 2016

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  6,   1;
  1,  7, 15,  10,   1;
  1,  9, 28,  35,  15,   1;
  1, 11, 45,  84,  70,  21,   1;
  1, 13, 66, 165, 210, 126,  28,  1;
  1, 15, 91, 286, 495, 462, 210, 36, 1; ...
...
(0, 1, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3,  1;
  0, 1, 5,  6,  1;
  0, 1, 7, 15, 10,  1;
  0, 1, 9, 28, 35, 15, 1. _Philippe Deléham_, Apr 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..495
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Molly Fenn and Eric Sommers, A transitivity result for ad-nilpotent ideals in type A, arXiv:2101.04091 [math.RT], 2021.
J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444 See Eq. 2.27. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004).

These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
Cf. A000108, A003558, A027989, A052535, A054142, A076756, A084938, A085478, A098435, A115139, A172431, A172991, A188648.
Cf. A001147, A001497.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n-k,k) ))); # G. C. Greubel, Aug 01 2019

[Binomial(2*n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T:=(n,k)->binomial(2*n-k,k): seq(seq(T(n,k), k=0..n), n=0..11);

Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] (* Emanuele Munarini, Apr 07 2011 *)

create_list(binomial(2*n-k,k),n,0,10,k,0,n); /* Emanuele Munarini, Apr 07 2011 */

T(n,k)=if(n<0,0,polcoeff(charpoly(matrix(n,n,i,j,-min(i,j))),k))

[[binomial(2*n-k,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f.: (1-t*z)/((1-t*z)^2-z). - Emeric Deutsch, May 31 2004
Column k has g.f.: (Sum_{j=0..k+1} binomial(k+1, 2j)*x^j)*x^k/(1-x)^(k+1). - Paul Barry, Jun 22 2005
Recurrence: T(n+2,k+2) = T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k). - Emanuele Munarini, Apr 07 2011
T(n, k) = binomial(2*n-k, k) = A085478(n, n-k), for n >= 0, k = 0..n. - Wolfdieter Lang, Mar 25 2020

A274895 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-1) (-1,-1) or (-2,0) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 12, 7, 6, 8, 36, 16, 14, 12, 16, 108, 37, 38, 26, 24, 32, 324, 86, 104, 84, 50, 48, 64, 972, 200, 290, 275, 192, 95, 96, 128, 2916, 465, 815, 913, 753, 436, 181, 192, 256, 8748, 1081, 2291, 3064, 3017, 2049, 990, 345, 384, 512, 26244, 2513, 6434, 10337

Offset: 1

R. H. Hardin, Jul 10 2016

Table starts
...1...1....2.....4......8......16.......32........64........128.........256
...2...4...12....36....108.....324......972......2916.......8748.......26244
...3...7...16....37.....86.....200......465......1081.......2513........5842
...6..14...38...104....290.....815.....2291......6434......18065.......50729
..12..26...84...275....913....3064....10337.....34921.....117975......398560
..24..50..192...753...3017...12217....49697....202749.....828828.....3391310
..48..95..436..2049...9863...48269...237807...1173787....5803040....28746995
..96.181..990..5602..32539..191974..1143185...6843349...41072451...246859250
.192.345.2253.15305.107369..767905..5539989..40156061..292253909..2133745005
.384.657.5121.41866.354366.3065418.26833885.236220817.2086382703.18485204565

			Some solutions for n=4 k=4
..0..1..0..2. .0..1..2..0. .0..1..0..2. .0..1..2..0. .0..1..2..1
..2..1..2..1. .1..2..0..1. .2..1..0..2. .1..2..0..1. .1..2..0..1
..1..0..2..1. .2..0..1..2. .1..0..2..1. .1..2..1..2. .1..0..1..2
..1..0..1..0. .2..0..1..0. .1..0..2..0. .2..0..1..2. .2..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A003945(n-2).
Column 2 is A052535(n+1).
Row 1 is A000079(n-2).
Row 2 is A003946(n-1).
Row 3 is A010912(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = a(n-1) +2*a(n-2) -a(n-4) for n>5
k=3: a(n) = a(n-1) +4*a(n-2) -6*a(n-4) -a(n-5) +4*a(n-6) -a(n-8) for n>10
k=4: [order 16] for n>18
k=5: [order 32] for n>34
k=6: [order 64] for n>66
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 3*a(n-1) for n>2
n=3: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3)
n=4: a(n) = 5*a(n-1) -9*a(n-2) +10*a(n-3) -6*a(n-4) +a(n-5) for n>6
n=5: [order 8] for n>9
n=6: [order 13] for n>14
n=7: [order 21] for n>22

A158943 INVERT transform of A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...).

Original entry on oeis.org

1, 1, 3, 5, 10, 19, 36, 69, 131, 250, 476, 907, 1728, 3292, 6272, 11949, 22765, 43371, 82629, 157422, 299915, 571388, 1088589, 2073943, 3951206, 7527704, 14341527, 27322992, 52054840, 99173120, 188941273, 359964521, 685792227, 1306548149

Offset: 1

Gary W. Adamson, Mar 31 2009

nonn

Equals row sums of triangle A158945.

Number of compositions of n into odd parts where there is 1 sort of part 1, 2 sorts of part 3, 3 sorts of part 5, ..., k sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

			The INVERT transform of (1, N, ...) begins (1, (N+1), ...) so that we have (1, 1, ...) placed in ascending magnitude in the bottom row. In the top row we place an equal number of descending terms: (..., 0, 3, 0, 2, 0, 1). Take the dot product of terms in top and bottom rows, adding the result to the next term A027656: (1, 0, 2, 0, 3, ...). a(6) = 19 given: 3, 0, 2, 0, 1 1, 1, 3, 5, 10 Dot product of top row terms * bottom row terms = (1, 0, 2, 0, 3) dot (1, 1, 3, 5, 10) = (3 + 0 + 6 + 0 + 10) = 19, which is added to the next term in (1, 0, 2, 0, 3, ...); i.e., (an 0) = 19.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A027656, A052535, A158944, A158945.

a:=[1,1,3,5];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jul 12 2019

I:=[1,1,3,5]; [n le 4 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 09 2019

A027656 := proc(n) if type(n,odd) then 0; else n/2+1 ; fi; end: L := [seq(A027656(n), n=0..100)] ; read("transforms"); INVERT(L) ; # R. J. Mathar, Apr 02 2009

LinearRecurrence[{1, 2, 0, -1}, {1, 1, 3, 5}, 40] (* Vincenzo Librandi, Jul 09 2019 *)

my(x='x+O('x^40)); Vec(x/(1-x-2*x^2+x^4)) \\ G. C. Greubel, Jul 12 2019

a=(x/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 12 2019

INVERT transform of (1, 0, 2, 0, 3, 0, 4, ...); i.e., the natural numbers interleaved with zeros.
From R. J. Mathar, Apr 02 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-4).
G.f.: x/(1 - x - 2*x^2 + x^4). (End)
The sequence is the second INVERT transform of (1, -1, 3, -5, 10, -19, ...). - Gary W. Adamson, Jul 08 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k). - Seiichi Manyama, Jun 12 2024

Extended by R. J. Mathar, Apr 02 2009

A275446 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {2,1,3,5,7,9,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 10, 6, 16, 4, 8, 30, 12, 11, 48, 36, 15, 82, 76, 8, 21, 128, 164, 32, 29, 204, 312, 112, 40, 312, 596, 288, 16, 55, 482, 1064, 704, 80, 76, 728, 1884, 1536, 320, 105, 1100, 3212, 3248, 960, 32, 145, 1640, 5428, 6464, 2624, 192

Offset: 0

Emeric Deutsch, Aug 17 2016

The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
number of entries in row n is 1+floor(n/3).
Sum of entries in row n is A052535(n).
T(n,0) = A103632(n+2).
Sum_{k>=0} k*T(n,k) = A275447(n).

			Row 4 is [3,4] because the compositions of 4 with parts in {2,1,3,5,7,...} are 22, 31, 13, 211, 121, 112, and 1111, having asymmetry degrees 0, 1, 1, 1, 0, 1, and 0, respectively.
Triangle starts:
  1;
  1;
  2;
  2,2;
  3,4;
  4,10;
  6,16,4.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A052535, A103632, A275447.

G := (1-z^4)/(1-z-z^2+(1-2*t)*z^3-z^4+z^6): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

Table[BinCounts[#, {0, 1 + Floor[n/3], 1}] &@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[ IntegerPartitions@ n, {_, a_, _} /; And[EvenQ@ a, a != 2]]], 1]]], {n, 0, 16}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

G.f.: G(t,z) = (1-z^4)/(1-z-z^2+(1-2*t)*z^3-z^4+z^6). In the more general situation of compositions into a[1]=1} z^(a[j]), we have G(t,z) = (1 + F(z))/(1 - F(z^2) - t*(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.

A373639 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-5k,k).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 16, 27, 43, 71, 119, 196, 322, 533, 882, 1455, 2402, 3970, 6558, 10829, 17887, 29548, 48804, 80608, 133146, 219925, 363254, 599998, 991044, 1636944, 2703794, 4465957, 7376591, 12184181, 20125051, 33241289, 54905857, 90690002, 149795989

Offset: 0

Seiichi Manyama, Jun 12 2024

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,0,-1).

Cf. A000930, A373640.

Cf. A052535, A122367, A373644.

LinearRecurrence[{1,0,2,0,0,-1},{1, 1, 1, 2, 4, 6},40] (* James C. McMahon, Jun 14 2024 *)
Table[Sum[Binomial[2n-5k,k],{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Sep 29 2024 *)

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

G.f.: 1 / (1 - x^3 - x/(1 - x^3)).

a(n) = a(n-1) + 2*a(n-3) - a(n-6) for n > 5.

A373638 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-5k,k).

Original entry on oeis.org

1, 1, 2, 5, 9, 21, 43, 92, 196, 414, 882, 1869, 3970, 8427, 17887, 37975, 80608, 171121, 363254, 771119, 1636944, 3474913, 7376591, 15659094, 33241289, 70564951, 149795989, 317988473, 675029164, 1432958824, 3041899638, 6457375642, 13707783053, 29099021980

Offset: 0

Seiichi Manyama, Jun 12 2024

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,0,-3,0,1).

Cf. A000045, A052535.

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

G.f.: 1 / (1 - x^2 - x/(1 - x^2)^2).

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5.

A052685 Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).

Original entry on oeis.org

1, 1, 4, 24, 168, 1680, 18720, 252000, 3830400, 65681280, 1251936000, 26225337600, 599710003200, 14851444608000, 396138155212800, 11320537003776000, 345079573622784000, 11176410365632512000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 633

Cf. A000142, A052535.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x^2)/(1-x-2*x^2+x^4) ))); // G. C. Greubel, Jun 03 2022

spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Prod(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

b[n_]:= b[n]= If[n<4, n!-2*Boole[n==3], b[n-1] +2*b[n-2] -b[n-4]]; (* b=A052535 *)
A052685[n_]:= n!*b[n];
Table[A052685[n], {n, 0, 40}] (* G. C. Greubel, Jun 03 2022 *)
With[{nn=20},CoefficientList[Series[(1-x^2)/(1-x-2 x^2+x^4),{x,0,nn}],x] Range[0,nn]!]

@CachedFunction
def b(n): # b = A052535
    if (n<4): return factorial(n) - 2*bool(n==3)
    else: return b(n-1) + 2*b(n-2) - b(n-4)
def A052685(n): return factorial(n)*b(n)
[A052685(n) for n in (0..40)] # G. C. Greubel, Jun 03 2022

E.g.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
D-finite recurrence: a(0)=1, a(1)=1, a(2)=4, a(3)=24, a(n) = n*a(n-1) + 2*n*(n-1)*a(n-2) - n*(n-1)*(n-2)*(n-3)*a(n-4).
a(n) = (n!/283)*Sum_{alpha=RootOf(1-z-2*Z^2+Z^4)} (27 + 112*alpha + 9*alpha^2 - 48*alpha^3)*alpha^(-1-n).
a(n) = n!*A052535(n). - R. J. Mathar, Nov 27 2011

A052969 Expansion of (1-x)/(1-x-2x^2+x^4).

Original entry on oeis.org

1, 0, 2, 2, 5, 9, 17, 33, 62, 119, 226, 431, 821, 1564, 2980, 5677, 10816, 20606, 39258, 74793, 142493, 271473, 517201, 985354, 1877263, 3576498, 6813823, 12981465, 24731848, 47118280, 89768153, 171023248, 325827706, 620755922, 1182643181

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1041
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A052535.

spec := [S,{S=Sequence(Prod(Union(Prod(Union(Sequence(Z),Z),Z),Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

CoefficientList[Series[(1-x)/(1-x-2x^2+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{1,2,0,-1},{1,0,2,2},40] (* Harvey P. Dale, Oct 20 2017 *)

G.f.: -(-1+x)/(1-2*x^2+x^4-x).
Recurrence: {a(0)=1, a(1)=0, a(2)=2, a(3)=2, a(n)-2*a(n+2)-a(n+3)+a(n+4)=0}.
Sum_(1/283*(29*_alpha+28*_alpha^3-76*_alpha^2+55)*_alpha^(-1-n), _alpha=RootOf(1-2*_Z^2+_Z^4-_Z)).
a(n)+a(n-1) = A052535(n). - R. J. Mathar, Nov 28 2011

More terms from James Sellers, Jun 05 2000

cp's OEIS Frontend

A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

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A274895 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-1) (-1,-1) or (-2,0) and new values introduced in order 0..2.

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A158943 INVERT transform of A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...).

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A275446 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {2,1,3,5,7,9,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).

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A373639 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k,k).

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A373638 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k).

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PARI

Formula

A052685 Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).

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A373639 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-5k,k).

A373638 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-5k,k).

A373644 a(n) = Sum_{k=0..floor(n/4)} binomial(2n-7k,k).