Gheorghe Coserea - cp's OEIS frontend

A321714 Numbers k such that lambda(k) = 12.

13, 26, 35, 39, 45, 52, 65, 70, 78, 90, 91, 104, 105, 112, 117, 130, 140, 144, 156, 180, 182, 195, 208, 210, 234, 260, 273, 280, 312, 315, 336, 360, 364, 390, 420, 455, 468, 520, 546, 560, 585, 624, 630, 720, 728, 780, 819, 840, 910, 936, 1008, 1040, 1092, 1170, 1260, 1365, 1456, 1560, 1638, 1680, 1820, 1872, 2184, 2340, 2520, 2730, 3120, 3276, 3640, 4095, 4368, 4680, 5040, 5460, 6552, 7280, 8190, 9360, 10920, 13104, 16380, 21840, 32760, 65520

Offset: 1

Gheorghe Coserea, Feb 21 2019

Here lambda is Carmichael's lambda function (see A002322).

R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.

Cf. A002322, A270562, A321713.

Select[Range[65520], CarmichaelLambda[#] == 12 &] (* Paolo Xausa, Feb 28 2024 *)

lambda(n) = { \\ A002322
  my(f=factor(n), fsz=matsize(f)[1]);
  lcm(vector(fsz, k, my(p=f[k,1], e=f[k,2]);
      if (p != 2, p^(e-1)*(p-1), e > 2, 2^(e-2), 2^(e-1))));
};
invlambda(n) = { \\ A270562
  if (n <= 0, return(0), n==1, return(2), n%2, return(0));
  my(f=factor(n), fsz=matsize(f)[1], g=1, h=1);
  for (k=1, fsz, my(p=f[k,1], e=1);
    while (n % lambda(p^e) == 0, e++); g *= p^(e-1));
  fordiv(n, d, if (isprime(d+1) && g % (d+1) != 0, h *= (d+1)));
  g *= h; if (lambda(g) != n, 0, g);
};
lambda_level(n) = {
  my(N = invlambda(n)); if (!N, return([])); my(s=List());
  fordiv(N, d, if (lambda(d) == n, listput(s, d)));
  Set(s);
};
lambda_level(12)

A321713 a(n) is the number of values k satisfying lambda(k)=n or zero if there is no solution, where lambda(k) is Carmichael's lambda function.

Original entry on oeis.org

2, 6, 0, 12, 0, 16, 0, 4, 0, 8, 0, 84, 0, 0, 0, 32, 0, 40, 0, 32, 0, 8, 0, 20, 0, 0, 0, 20, 0, 64, 0, 8, 0, 0, 0, 480, 0, 0, 0, 80, 0, 48, 0, 12, 0, 8, 0, 160, 0, 0, 0, 20, 0, 16, 0, 4, 0, 8, 0, 1216, 0, 0, 0, 8, 0, 64, 0, 0, 0, 16, 0, 872, 0, 0, 0, 0, 0, 24, 0, 160, 0, 8, 0, 532, 0, 0, 0, 52, 0, 120, 0, 12, 0, 0, 0, 424, 0, 0, 0, 100

Offset: 1

Gheorghe Coserea, Feb 21 2019

nonn

			For n=12 there are a(12)=84 values N satisfying lambda(N)=12; the values are enumerated in A321714.

Bertram Felgenhauer, Table of n, a(n) for n = 1..10000 (first 3023 terms by Gheorghe Coserea)
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.

Cf. A002322, A270562, A321714.

lambda(n) = { \\ A002322
  my(f=factor(n), fsz=matsize(f)[1]);
  lcm(vector(fsz, k, my(p=f[k,1], e=f[k,2]);
      if (p != 2, p^(e-1)*(p-1), e > 2, 2^(e-2), 2^(e-1))));
};
invlambda(n) = { \\ A270562
  if (n <= 0, return(0), n==1, return(2), n%2, return(0));
  my(f=factor(n), fsz=matsize(f)[1], g=1, h=1);
  for (k=1, fsz, my(p=f[k,1], e=1);
    while (n % lambda(p^e) == 0, e++); g *= p^(e-1));
  fordiv(n, d, if (isprime(d+1) && g % (d+1) != 0, h *= (d+1)));
  g *= h; if (lambda(g) != n, 0, g);
};
lambda_level(n) = {
  my(N = invlambda(n)); if (!N, return([])); my(s=List());
  fordiv(N, d, if (lambda(d) == n, listput(s, d)));
  Set(s);
};
a(n) = length(lambda_level(n));
vector(100, n, a(n))

b(n) = { \\ number of k satisfying lambda(k) | n
my(R = 1);
fordiv (n, d, if(isprime(d+1),
  my(e = 1); while(n % (d+1) == 0, n /= d+1; e++);
  if (d == 1 && e > 1, e++); R *= e+1));
R
};
a(n) = if (n <= 0, 0, n == 1, 2, n % 2, 0, sumdiv(n, d, moebius(n/d) * b(d)));
vector(100, n, a(n)) \\ Bertram Felgenhauer, Mar 27 2022

A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 0, 3, 0, 0, 11, 9, 0, 1, 53, 120, 60, 40, 9, 309, 1410, 1800, 1590, 885, 216, 2119, 16560, 39960, 55120, 52065, 29016, 7570, 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435, 148329, 2624496, 15606360, 48387024, 99650670, 141429456, 135382464, 79738800, 22040361, 1468457, 36080100, 304274880, 1323453180, 3760709526, 7493549868, 10570597800, 10199809980, 6103007505, 1721632024

Offset: 0

Gheorghe Coserea, Nov 27 2018

			For n=3 we have s1 = z1 + z2 + z3, s2 = z1^2 + z2^2 + z3^2, s12 = z1*z2 + z1*z3 + z2*z3, f1 = z1^2 + z2^2 + z3^2 + t*z2*z3 + z1*(z2 + z3), f2 = z1^2 + z2^2 + z3^2 + t*z1*z3 + z2*(z1 + z3), f3 = z1^2 + z2^2 + z3^2 + t*z1*z2 + z3*(z1 + z2), [(z1*z2*z3)^2] f1*f2*f3 = 11 + 9*t + t^3, therefore P_3(t) = 11 + 9*t + t^3.
A(x;t) = 1 + x + 3*x^2 + (11 + 9*t + t^3)*x^3 + (53 + 120*t + 60*t^2 + 40*t^3 + 9*t^4)*x^4 + ...
Triangle starts:
n\k [0]    [1]     [2]     [3]      [4]      [5]      [6]      [7]
[0] 1;
[1] 1;     0;
[2] 3;     0;      0;
[3] 11,    9,      0,      1;
[4] 53,    120,    60,     40,      9;
[5] 309,   1410,   1800,   1590,    885,     216;
[6] 2119,  16560,  39960,  55120,   52065,   29016,   7570;
[7] 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435;
[8] ...

Gheorghe Coserea, Rows n = 0..13, flattened
Shmuel Friedland, Giorgio Ottaviani, The number of singular vector tuples and uniqueness of best rank one approximation of tensors, arXiv:1210.8316 [math.AG], 2013.

Cf. A000255, A000681, A007107, A274308, A284989.

P(n, t='t) = {
  my(z=vector(n, k, eval(Str("z", k))),
     s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
     f=vector(n, k, s2 + t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
  for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
  for (k=1, n, g=polcoef(g, 2, z[k]));
  g;
};
seq(N) = concat([[1], [1, 0], [3, 0, 0]], apply(n->Vecrev(P(n,'t)), [3..N]));
concat(seq(9))

Let z1..zn be n variables and s1 = Sum_{k=1..n} zk, s2 = Sum_{k=1..n} zk^2, s12 = (s1^2 - s2)/2, fk = s2 + t*(s12 - zk*(s1 - zk)) + zk*(s1 - zk) for k=1..n; we define P_n(t) = [(z1..zn)^2] Product_{k=1..n} fk.
A000255(n) = T(n,0).
A007107(n) = T(n,n).
A000681(n) = Sum_{k=0..n} T(n,k).
A274308(n) = Sum_{k=0..n} T(n,k)*2^k.

A321709 Number of genus 9 rooted hypermaps with n darts.

Original entry on oeis.org

640237370572800, 125042114810880000, 12663309118470912000, 885531242655070387200, 48050907406540671926016, 2155959161260047868965120, 83244478130310272277600000, 2842577637658581528366266112, 87562415111185051655245105920, 2469899876154506406497571909952

Offset: 19

Gheorghe Coserea, Nov 17 2018

nonn

Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 9 of A321710.

A318104 Number of genus 4 rooted hypermaps with n darts.

Original entry on oeis.org

8064, 579744, 23235300, 684173164, 16497874380, 344901105444, 6471056247920, 111480953909328, 1792031518697232, 27197316623478960, 393207192141924744, 5453210050430783640, 72949244341257096792, 945523594111460363208, 11918067649004916470640, 146538779626167833263888, 1762112462707129510538640

Offset: 9

Gheorghe Coserea, Nov 12 2018

nonn

Column k = 4 of A321710.

a(n) = 0 for n < 9. - N. J. A. Sloane, Dec 24 2018

			A(x) = 8064*x^9 + 579744*x^10 + 23235300*x^11 + 684173164*x^12 + ...

Gheorghe Coserea, Table of n, a(n) for n = 9..109
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 6
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Cf. A000257, A118093, A214817, A214818, A321710.

y = (1 - Sqrt[1 - 8 x])/(4 x);
gf = -y (y-1)^9 (262 y^14 - 4716 y^13 + 78327 y^12 - 569134 y^11 + 3266910 y^10 - 12675726 y^9 + 37548087 y^8 - 82680972 y^7 + 137674842 y^6 - 170295272 y^5 + 152918277 y^4 - 94811622 y^3 + 37127810 y^2 - 7566846 y + 505869)/(4 (y-2)^17 (y+1)^13);
Drop[CoefficientList[gf + O[x]^26, x], 9] (* Jean-François Alcover, Feb 07 2019, from PARI *)

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(-y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13));
};
seq(17)

G.f.: -y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13), where y = C(2*x), C being the g.f. for A000108.

A321708 Number of genus 8 rooted hypermaps with n darts.

Original entry on oeis.org

2324754432000, 392578737868800, 34628437757170944, 2123123613378727680, 101617019078765734656, 4043718573909964925184, 139173710438785042157056, 4255877128208539192614720, 117904584208478764267020480, 3002991800410924564910152160, 71123951019277304732006170020

Offset: 17

Gheorghe Coserea, Nov 17 2018

nonn

Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 8 of A321710.

A321707 Number of genus 7 rooted hypermaps with n darts.

Original entry on oeis.org

10897286400, 1560315052800, 117805728533760, 6234567636407040, 259518044572234560, 9042873557130178560, 274213957041780607040, 7429773263737371426000, 183321599847270732775284, 4178263675886440605897708, 88944569813776527012125700, 1783998282211666387087167804

Offset: 15

Gheorghe Coserea, Nov 17 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 15..115
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 7 of A321710.

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(y*(y - 1)^15*(65934428*y^29 - 2373639408*y^28 + 78065287272*y^27 - 1239982167455*y^26 + 15824435903631*y^25 - 147973440711114*y^24 + 1121119206347992*y^23 - 6897821674771866*y^22 + 35411504747483046*y^21 - 153195355715747532*y^20 + 564430051160248776*y^19 - 1782487694948370405*y^18 + 4847762532445875077*y^17 - 11385796789588847190*y^16 + 23121306258262005552*y^15 - 40583304876425900476*y^14 + 61456131914028384816*y^13 - 80013191284851054552*y^12 + 89093563965211655640*y^11 - 84217909084706439705*y^10 + 66908561791095714729*y^9 - 44081770356535124534*y^8 + 23656775380308201720*y^7 - 10093706157332984394*y^6 + 3311499493215102574*y^5 - 796565568171137388*y^4 + 130914104820311544*y^3 - 13151257876934851*y^2 + 664822027105443*y - 11057829184170)/(2*(y - 2)^32*(y + 1)^25));
};
seq(12)

G.f.: y*(y - 1)^15*(65934428*y^29 - 2373639408*y^28 + 78065287272*y^27 - 1239982167455*y^26 + 15824435903631*y^25 - 147973440711114*y^24 + 1121119206347992*y^23 - 6897821674771866*y^22 + 35411504747483046*y^21 - 153195355715747532*y^20 + 564430051160248776*y^19 - 1782487694948370405*y^18 + 4847762532445875077*y^17 - 11385796789588847190*y^16 + 23121306258262005552*y^15 - 40583304876425900476*y^14 + 61456131914028384816*y^13 - 80013191284851054552*y^12 + 89093563965211655640*y^11 - 84217909084706439705*y^10 + 66908561791095714729*y^9 - 44081770356535124534*y^8 + 23656775380308201720*y^7 - 10093706157332984394*y^6 + 3311499493215102574*y^5 - 796565568171137388*y^4 + 130914104820311544*y^3 - 13151257876934851*y^2 + 664822027105443*y - 11057829184170)/(2*(y - 2)^32*(y + 1)^25), where y=A000108(2*x).

A321706 Number of genus 6 rooted hypermaps with n darts.

Original entry on oeis.org

68428800, 8099018496, 511859777472, 22925949056640, 815521082030784, 24494440792190400, 645212095792089220, 15292175926873102956, 332150183310464271324, 6702637985834037183508, 126995200843857803023176, 2278149500006567629947864, 38954050134978747926573016

Offset: 13

Gheorghe Coserea, Nov 17 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 13..113
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 8
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 6 of A321710.

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(-y*(y - 1)^13*(1080091*y^24 - 32402730*y^23 + 889296813*y^22 - 11575684382*y^21 + 120636055215*y^20 - 908735922846*y^19 + 5491340556019*y^18 - 26587756725282*y^17 + 105914199493428*y^16 - 349844034215428*y^15 + 966356094916770*y^14 - 2240740995310188*y^13 + 4368032453176430*y^12 - 7149882085566108*y^11 + 9789363335577126*y^10 - 11134972065337540*y^9 + 10413235525450707*y^8 - 7887398782084338*y^7 + 4736927774219617*y^6 - 2188131419800854*y^5 + 743586620967027*y^4 - 173682661266854*y^3 + 24974862235959*y^2 - 1816988020602*y + 43470403150)/(4*(y - 2)^27*(y + 1)^21));
};
seq(13)

G.f.: -y*(y - 1)^13*(1080091*y^24 - 32402730*y^23 + 889296813*y^22 - 11575684382*y^21 + 120636055215*y^20 - 908735922846*y^19 + 5491340556019*y^18 - 26587756725282*y^17 + 105914199493428*y^16 - 349844034215428*y^15 + 966356094916770*y^14 - 2240740995310188*y^13 + 4368032453176430*y^12 - 7149882085566108*y^11 + 9789363335577126*y^10 - 11134972065337540*y^9 + 10413235525450707*y^8 - 7887398782084338*y^7 + 4736927774219617*y^6 - 2188131419800854*y^5 + 743586620967027*y^4 - 173682661266854*y^3 + 24974862235959*y^2 - 1816988020602*y + 43470403150)/(4*(y - 2)^27*(y + 1)^21), where y=A000108(2*x).

A321705 Number of genus 5 rooted hypermaps with n darts.

Original entry on oeis.org

604800, 57170880, 2936606400, 108502598960, 3225186125460, 81861294718764, 1840409325096500, 37558997857897164, 708015469597497732, 12488421105878928700, 208161512148250424484, 3304395638081490531324, 50267199680265668419244, 736516493829967530909204, 10437808798822929984593100

Offset: 11

Gheorghe Coserea, Nov 17 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 11..111
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 7
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 5 of A321710.

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(y*(y - 1)^11*(13150*y^19 - 315600*y^18 + 6947865*y^17 - 70489470*y^16 + 569637816*y^15 - 3253135788*y^14 + 14658702716*y^13 - 51696766668*y^12 + 146255446788*y^11 - 332779761068*y^10 + 610739916966*y^9 - 900544355928*y^8 + 1057440629016*y^7 - 973453624356*y^6 + 685359139356*y^5 - 355019010868*y^4 + 127180243662*y^3 - 28342783668*y^2 + 3224985513*y - 120590634)/(4*(y - 2)^22*(y + 1)^17));
};
seq(15)

G.f.: y*(y - 1)^11*(13150*y^19 - 315600*y^18 + 6947865*y^17 - 70489470*y^16 + 569637816*y^15 - 3253135788*y^14 + 14658702716*y^13 - 51696766668*y^12 + 146255446788*y^11 - 332779761068*y^10 + 610739916966*y^9 - 900544355928*y^8 + 1057440629016*y^7 - 973453624356*y^6 + 685359139356*y^5 - 355019010868*y^4 + 127180243662*y^3 - 28342783668*y^2 + 3224985513*y - 120590634)/(4*(y - 2)^22*(y + 1)^17), where y=A000108(2*x).

A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

Original entry on oeis.org

1, 3, 12, 1, 56, 15, 288, 165, 8, 1584, 1611, 252, 9152, 14805, 4956, 180, 54912, 131307, 77992, 9132, 339456, 1138261, 1074564, 268980, 8064, 2149888, 9713835, 13545216, 6010220, 579744, 13891584, 81968469, 160174960, 112868844, 23235300, 604800, 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880, 608583680, 5702382933, 19588944336, 28540603884, 16497874380, 2936606400, 68428800, 4107939840, 47168678571, 206254571236, 404562365316, 344901105444, 108502598960, 8099018496

Offset: 1

Gheorghe Coserea, Nov 17 2018

Row n contains floor((n+1)/2) = A008619(n-1) terms.

			Triangle starts:
n\k  [0]       [1]        [2]         [3]         [4]        [5]
[1]  1;
[2]  3;
[3]  12,       1;
[4]  56,       15;
[5]  288,      165,       8;
[6]  1584,     1611,      252;
[7]  9152,     14805,     4956,       180;
[8]  54912,    131307,    77992,      9132;
[9]  339456,   1138261,   1074564,    268980,     8064;
[10] 2149888,  9713835,   13545216,   6010220,    579744;
[11] 13891584, 81968469,  160174960,  112868844,  23235300,  604800;
[12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
[13] ...

Gheorghe Coserea, Rows n = 1..42, flattened
Alain Giorgetti and Timothy R. S. Walsh, Enumeration of hypermaps of a given genus, Ars Math. Contemp. 15 (2018) 225-266.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Columns k=0..9 give: A000257 (k=0), A118093 (k=1), A214817 (k=2), A214818 (k=3), A318104 (k=4), A321705 (k=5), A321706 (k=6), A321707 (k=7), A321708 (k=8), A321709 (k=9).
Row sums give A003319(n+1).
Cf. A008619, A060593.

l1[f_,n_] := Sum[(i-1)t[i]D[f,t[i-1]], {i,2,n}];
m1[f_,n_] := Sum[(i-1)t[j]t[i-j]D[f,t[i-1]] + j(i-j)t[i+1]D[f,t[j],t[i-j]], {i,2,n},{j,i-1}];
ff[1] = x^2 t[1];
ff[n_] := ff[n] = Simplify@(2x l1[ff[n-1],n] + m1[ff[n-1],n] + Sum[t[i+1]j(i-j)D[ff[k],t[j]]D[ff[n-1-k],t[i-j]], {i,2,n-1},{j,i-1},{k,n-2}]) / n;
row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;;2]][[;;Quotient[n+1,2]]];
Table[row[n], {n,14}] (* Andrei Zabolotskii, Jun 27 2025, after the PARI code *)

L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
M1(f, N) = {
  sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
      j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
};
F(N) = {
  my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
  f[1] = u*v*t[1];
  for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
    sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
    j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
    f[n] /= n);
  f;
};
seq(N) = {
  my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
     g=vector(#v, n, Polrev(Vec(n * v[n]))));
  apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
};
concat(seq(14))

A000257(n)=T(n,0), A118093(n)=T(n,1), A214817(n)=T(n,2), A214818(n)=T(n,3), A060593(n)=T(2*n+1,n)=(2*n)!/(n+1), A003319(n+1)=Sum_{k=0..floor((n-1)/2)} T(n,k).

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User: Gheorghe Coserea

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