A000211 -id:A000211 - cp's OEIS frontend

A000440 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-4 places.

9, 30, 180, 980, 8326, 70272, 695690, 7518720, 89193276, 1148241458, 15947668065, 237613988040, 3780133322620, 63945806121448, 1146081593303784, 21693271558730304, 432411684714253605, 9053476937543082240, 198641103956454088919

Offset: 4

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000492, A000476, A000380, A000388.

Snkgf := (t, u) - >(1 - t*u)^( - 1)*(1 - (1 + 2*t)*u - t*u^2 + t^3*u^3)^( - 1); sigmankgf := (t, u) - >(1 - 2*t^2*u^2 - 2*t^2*(1 + t)*u^3 + 3*t^4*u^4)*Snkgf(t, u); f := (n, k) - >coeff(sum(coeff(subs(u=0, diff(sigmankgf(t, u), u$n))/n!, t, j)*(n - j)!*(y - 1)^j, j =0..n), y, k); seq(f(i, 4), i=4..30); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1+t) u^3 + 3 t^4 (u^4)) (1 - t*u)^(-1) (1 - (1+2t) u - t*u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; f[n_, k_] := Coefficient[Sum[Coefficient[ds[t, n]/n!, t, j]*(n-j)!*(y-1)^j, {j, 0, n}], y, k]; Table[f[i, 4], {i, 4, 22}] (* Jean-François Alcover, May 27 2011, after Maple prog. *)

a(n) = coefficient of y^4 in sum_0^n sigma_{n, k}(n-k)!(y-1)^k on y where the sigma_{n, k} have generating function sigma(t, u)=(1-2t^2(u^2)-2t^2(1+t)u^3+3t^4(u^4))(1-tu)^(-1)(1-(1+2t)u-tu^2+t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000476 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-1 places.

Original entry on oeis.org

15, 72, 609, 4960, 46188, 471660, 5275941, 64146768, 842803767, 11902900380, 179857257960, 2895705788736, 49491631601635, 895010868095256, 17074867330880805, 342733960299356800, 7220616209235766260, 159312370008282356844, 3673720238903201471593

Offset: 5

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000440, A000492, A000380, A000388.

seq(f(n,1), n=5..30); # where code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1 + t) u^3 + 3 t^4 (u^4)) (1 - t* u)^(-1) (1 - (1 + 2 t) u - t *u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; su[n_] := su[n] = Sum[ Coefficient[ds[t, n]/n!, t, j]*(n - j)!*(y - 1)^j, {j, 0, n}]; f[n_, k_] := Coefficient[su[n], y, k]; Table[f[n, 1], {n, 5, 23}] (* Jean-François Alcover, Sep 01 2011, after Maple prog. *)

a(n) = coefficient of y in sum_0^n sigma_{n, k}(n-k)!(y-1)^k on y where the sigma_{n, k} have generating function sigma(t, u)=(1-2t^2(u^2)-2t^2(1+t)u^3+3t^4(u^4))(1-tu)^(-1)(1-(1+2t)u-tu^2+t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000470 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-5 places.

Original entry on oeis.org

13, 72, 595, 4096, 39078, 379760, 4181826, 49916448, 647070333, 9035216428, 135236990388, 2159812592384, 36658601139066, 658942295734944, 12504663617290908, 249823152134646144, 5241223014084306270, 115206851288747267148, 2647678812396326064043

Offset: 5

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000492, A000440, A000476, A000380, A000388.

seq(f(n,5), n=5..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1 + t) u^3 + 3 t^4 (u^4)) (1 - t* u)^(-1) (1 - (1 + 2 t) u - t *u^2 + t^3 (u^3))^(-1);ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; su[n_] := su[n] = Sum[ Coefficient[ds[t, n]/n!, t, j]*(n - j)!*(y - 1)^j, {j, 0, n}]; f[n_, k_] := Coefficient[su[n], y, k]; Table[f[n, 5], {n, 5, 23}] (* Jean-François Alcover, Sep 01 2011, after Maple prog. *)

a(n) = coefficient of y^5 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000492 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.

Original entry on oeis.org

20, 154, 1676, 14292, 155690, 1731708, 21264624, 280260864, 3970116255, 60113625680, 969368687752, 16588175089420, 300272980075896, 5733025551810600, 115148956467702600, 2427199940533198992, 53576182138937428377, 1235917889588345408586

Offset: 6

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000440, A000476, A000380, A000388.

seq(f(n,6), n=6..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1-2t^2 (u^2) - 2t^2 (1+t) u^3 + 3t^4 (u^4)) (1-t*u)^(-1) (1-(1+2t)u - t*u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; f[n_, k_] := Coefficient[Sum[ Coefficient[ ds[t, n]/n!, t, j]*(n-j)!*(y-1)^j, {j, 0, n}], y, k]; a[n_] := f[n, 6]; Table[a[n], {n, 6, 25}] (* Jean-François Alcover, Feb 09 2016 *)

a(n) = coefficient of y^6 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000500 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.

Original entry on oeis.org

31, 304, 4230, 43880, 547338, 6924960, 94714620, 1375878816, 21273204330, 348919244768, 6056244249682, 110955673493568, 2140465858763844, 43379533256972640, 921616584567907176, 20485188316420940640, 475499882089797554181, 11506280235885243825696

Offset: 7

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000492, A000470, A000440, A000476, A000380, A000388.

seq(f(n,7), n=7..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] := (1-2*t^2*u^2 - 2*t^2*(1+t)*u^3 + 3*t^4*u^4)/((1-t*u)*(1-(1 + 2*t)*u - t*u^2 + t^3*u^3)); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; f[n_, k_] := Coefficient[ Sum[ Coefficient[ds[t, n]/n!, t, j]*(n-j)!*(y-1)^j, {j, 0, n}], y, k]; a[n_] := f[n, 7]; Table[Print[an = a[n]]; an, {n, 7, 24}] (* Jean-François Alcover, Jan 25 2013, after Maple code *)

a(n) = coefficient of y^7 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A176212 Terms of A176211, duplicates removed.

Original entry on oeis.org

6, 9, 13, 20, 31, 36, 49, 54, 78, 81, 117, 120, 125, 169, 180, 186, 201, 216, 260, 279, 294, 324, 400, 403, 441, 468, 486, 523, 620, 637, 702, 720, 729, 750, 845, 961, 980, 1014, 1053, 1080, 1116, 1125, 1206, 1296, 1366, 1519, 1521, 1560, 1620, 1625, 1674, 1764, 1809, 1944, 2197

Offset: 1

Vladimir Shevelev, Apr 12 2010

nonn

The terms are permanents of a set of certain symmetric (0,1)-matrices as detailed in A176211. Thus the sequence solves a symmetric version of Gristein problem: to find all the values of permanent of all square (0,1) matrices, which contain exactly three 1's in each row and column (see the list of unsolved problems in chapter 8 of Minc's book).

H. Minc, Permanents, Addison-Wesley, 1978.

V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317

Cf. A176211, A176210, A000211.

f(n) = fibonacci(n+1) + fibonacci(n-1) + 2; \\ A000211
lista(nn) = {my(v = vector(nn, k, f(k+2))); my(vmax = vecmax(v)); my(w =  vector(nn, k, [0, logint(vmax, v[k])])); my(list=List()); forvec(x = w, if (vecmax(x), my(y = prod(k=1, #v, v[k]^x[k])); if (y <= vmax, listput(list, y)););); Vec(vecsort(list,,8));}
lista(14) \\ Michel Marcus, Jan 06 2021

A001925 From rook polynomials.

Original entry on oeis.org

1, 6, 22, 64, 162, 374, 809, 1668, 3316, 6408, 12108, 22468, 41081, 74202, 132666, 235160, 413790, 723530, 1258225, 2177640, 3753096, 6444336, 11028792, 18818664, 32024977, 54367374, 92094334, 155688208, 262711866, 442556798, 744355673, 1250157228

Offset: 0

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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 = 0..1000
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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (5,-8,2,6,-4,-1,1).

Cf. A002940.

A001925:=-(1+z)/(z**2+z-1)**2/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

nn = 40; CoefficientList[Series[(1 + x)/((1 - x - x^2)^2*(1 - x)^3), {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{5,-8,2,6,-4,-1,1},{1,6,22,64,162,374,809},40] (* Harvey P. Dale, Oct 15 2021 *)

Riordan gives the g.f. (1+x)/[(1-x-x^2)^2*(1-x)^3].

A061702 Triangle T(n,k) defined by Sum_{n >= 0,m >= 0} T(n,m)x^my^n = 1 + y(1 + 3x - 4x^2y - 3x^2y^2 - 3x^3y^2 + 4x^4y^3)/((1 - y - 2xy - xy^2 + x^3y^3)(1 - xy)).

Original entry on oeis.org

1, 1, 3, 1, 6, 5, 1, 9, 18, 6, 1, 12, 42, 44, 9, 1, 15, 75, 145, 95, 13, 1, 18, 117, 336, 420, 192, 20, 1, 21, 168, 644, 1225, 1085, 371, 31, 1, 24, 228, 1096, 2834, 3880, 2588, 696, 49, 1, 27, 297, 1719, 5652, 10656, 11097, 5823, 1278, 78, 1, 30, 375, 2540, 10165

Offset: 0

Vladeta Jovovic, Jun 18 2001

It is uncertain if the initial term should be 0 or 1. Both make sense. I have changed the data line to start with 1, in agreement with Riordan (1954). - N. J. A. Sloane, Jun 28 2015

See Riordan 1954 page 21 equation (24). - Michael Somos, Aug 26 2015

			Triangle begins:
1,
1,3,
1,6,5,
1,9,18,6,
1,12,42,44,9,
1,15,75,145,95,13,
1,18,117,336,420,192,20,
1,21,168,644,1225,1085,371,31,
1,24,228,1096,2834,3880,2588,696,49,
1,27,297,1719,5652,10656,11097,5823,1278,78,
1,30,375,2540,10165,24626,35045,29380,12535,2310,125,
... (from _N. J. A. Sloane_, Jun 28 2015)
Sum_{n, k} T(n, k) u^n t^k = 1 + (1 + 3*t)*u + (1 + 6*t + 5*t^2)*u^2 + ...

R. P. Stanley, Enumerative Combinatorics I, Example 4.7.17.

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] See Table 1.

Cf. A000183, row sums: A061703, third column: A000338, fourth column: A000561, fifth column: A000562, sixth column: A000563, seventh column: A000564, eighth column: A000565.

max = 11; f[x_, y_] := 1 + y*(1 + 3*x - 4*x^2*y - 3*x^2*y^2 - 3*x^3*y^2 + 4*x^4*y^3)/((1 - y - 2*x*y - x*y^2 + x^3*y^3)*(1 - x*y)); se = Series[f[x, y], {x, 0, max}, {y, 0, max}]; coes = CoefficientList[se, {x, y}] ; t[n_, k_] := coes[[k, n]]; Flatten[ Table[t[n, k], {n, 1, max}, {k, 1, n}]](* Jean-François Alcover, Oct 24 2011 *)

Edited by N. J. A. Sloane, Jun 28 2015

A000338 Expansion of x^3(5-2x)*(1-x^3)/(1-x)^4.

Original entry on oeis.org

5, 18, 42, 75, 117, 168, 228, 297, 375, 462, 558, 663, 777, 900, 1032, 1173, 1323, 1482, 1650, 1827, 2013, 2208, 2412, 2625, 2847, 3078, 3318, 3567, 3825, 4092, 4368, 4653, 4947, 5250, 5562, 5883, 6213, 6552, 6900, 7257, 7623, 7998, 8382, 8775, 9177, 9588, 10008, 10437, 10875, 11322, 11778

Offset: 3

N. J. A. Sloane

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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 = 3..1000
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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

ff := n->9/2*n^2-15/2*n; seq(ff(n), n=3..60); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001, sequence without a(3).

nn = 100; CoefficientList[Series[(5 - 2 x) (1 - x^3)/(1 - x)^4, {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
LinearRecurrence[{3,-3,1},{5,18,42,75},60] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = 3*A095794(n-2), n>3. - R. J. Mathar, May 30 2022
G.f.: (1+x+x^2)*(5-2*x)*x^3/(1-x)^3. - Simon Plouffe in his 1992 dissertation
Sum_{n>=3} 1/a(n) = log(3)/5 + Pi*sqrt(3)/45 = 0.3406424... - R. J. Mathar, Apr 22 2024
a(n) = 5*A005448(n-2) -2*A005448(n-3). - R. J. Mathar, Apr 22 2024

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000561 Number of discordant permutations.

Original entry on oeis.org

6, 44, 145, 336, 644, 1096, 1719, 2540, 3586, 4884, 6461, 8344, 10560, 13136, 16099, 19476, 23294, 27580, 32361, 37664, 43516, 49944, 56975, 64636, 72954, 81956, 91669, 102120, 113336, 125344, 138171, 151844, 166390, 181836, 198209, 215536, 233844, 253160, 273511, 294924, 317426, 341044

Offset: 3

N. J. A. Sloane

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

G. C. Greubel, Table of n, a(n) for n = 3..1000
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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(9/2)*n^3-(45/2)*n^2+29*n: n in [3..45]]; // Vincenzo Librandi, Feb 10 2016

f := n->9/2*n^3-45/2*n^2+29*n; seq(f(n), n=0..50); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
A000561:=-(-6-20*z-5*z**2+4*z**3)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{4, -6, 4, -1}, {6, 44, 145, 336}, 50] (* Jean-François Alcover, Feb 10 2016 *)
Drop[CoefficientList[Series[x^3(6+20x+5x^2-4x^3)/(1-x)^4,{x,0,50}],x],3] (* Harvey P. Dale, Jul 20 2021 *)

for(n=3, 45, print1(n*(9*n^2 - 45*n + 58)/2, ", ")) \\ G. C. Greubel, Nov 23 2018

[n*(9*n^2 - 45*n + 58)/2 for n in (3..45)] # G. C. Greubel, Nov 23 2018

G.f.: x^3*(6 + 20*x + 5*x^2 - 4*x^3) / (1 - x)^4. - Jeffrey Shallit [adapted by Vincenzo Librandi, Feb 10 2016]
a(n) = n*(9*n^2 - 45*n + 58)/2. - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
E.g.f.: x*(-22 - 4*x + (22 - 18*x + 9*x^2)*exp(x))/2. - G. C. Greubel, Nov 23 2018

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

cp's OEIS Frontend

A000440 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-4 places.

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A000476 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-1 places.

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A000470 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-5 places.

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A000492 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.

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A000500 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.

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A176212 Terms of A176211, duplicates removed.

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A001925 From rook polynomials.

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A061702 Triangle T(n,k) defined by Sum_{n >= 0,m >= 0} T(n,m)x^my^n = 1 + y(1 + 3x - 4x^2y - 3x^2y^2 - 3x^3y^2 + 4x^4y^3)/((1 - y - 2xy - xy^2 + x^3y^3)(1 - xy)).

A000338 Expansion of x^3(5-2x)*(1-x^3)/(1-x)^4.