cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A385910 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y), read by rows.

Original entry on oeis.org

1, 0, 1, 0, -1, 3, 0, 1, -8, 12, 0, 1, 7, -49, 55, 0, 0, 9, 56, -296, 273, 0, -2, 9, 14, 498, -1815, 1428, 0, 0, -23, 91, -288, 4181, -11284, 7752, 0, -1, -3, -108, 522, -4487, 33168, -70924, 43263, 0, 1, -23, 82, -579, 3971, -49239, 253590, -449616, 246675, 0, 0, 5, -373, 2419, -6510, 46017, -478291, 1892593, -2869779, 1430715
Offset: 1

Views

Author

Paul D. Hanna, Jul 14 2025

Keywords

Comments

Row sums form the Catalan numbers (A000108), with g.f. C(x) = 1 + x*C(x)^2.
Main diagonal equals A001764, with g.f. D(x) = 1 + x*D(x)^3.
Column 1 equals A385911.

Examples

			G.f. A(x,y) = x + y*x^2 + (3*y^2 - y)*x^3 + (12*y^3 - 8*y^2 + y)*x^4 + (55*y^4 - 49*y^3 + 7*y^2 + y)*x^5 + (273*y^5 - 296*y^4 + 56*y^3 + 9*y^2)*x^6 + (1428*y^6 - 1815*y^5 + 498*y^4 + 14*y^3 + 9*y^2 - 2*y)*x^7 + (7752*y^7 - 11284*y^6 + 4181*y^5 - 288*y^4 + 91*y^3 - 23*y^2)*x^8 + (43263*y^8 - 70924*y^7 + 33168*y^6 - 4487*y^5 + 522*y^4 - 108*y^3 - 3*y^2 - y)*x^9 + (246675*y^9 - 449616*y^8 + 253590*y^7 - 49239*y^6 + 3971*y^5 - 579*y^4 + 82*y^3 - 23*y^2 + y)*x^10 + ...
where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
TRIANGLE.
Triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
  1;
  0,  1;
  0, -1,   3;
  0,  1,  -8,   12;
  0,  1,   7,  -49,    55;
  0,  0,   9,   56,  -296,   273;
  0, -2,   9,   14,   498, -1815,   1428;
  0,  0, -23,   91,  -288,  4181, -11284,    7752;
  0, -1,  -3, -108,   522, -4487,  33168,  -70924,    43263;
  0,  1, -23,   82,  -579,  3971, -49239,  253590,  -449616,   246675;
  0,  0,   5, -373,  2419, -6510,  46017, -478291,  1892593, -2869779,   1430715;
  0,  0,  -2,  -65, -3746, 28523, -74367,  554792, -4334344, 13891755, -18418400, 8414640; ...

Links

Crossrefs

Cf. A384830 (y=-1), A376226 (y=2), A385911 (column 1), A385912 (a diagonal).
Cf. A000108 (row sums), A001764 (main diagonal).

Programs

  • PARI
    {T(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 + 3*y*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 2*y*x*Ax^2 ), #A+1)); A[n+1]}
\\ Print the rows of the triangle
my(Rown); for(n=1, 12, Rown = T(n); for(k=0,n-1, print1(polcoef(Rown,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies the following formulas.
(1) A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
(2) A(x,y=1) = C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) T(n+1,n) = A001764(n) for n >= 0, with g.f. D(x) = 1 + x*D(x)^3.

A384270 G.f. satisfies A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3).

Original entry on oeis.org

1, 1, 4, 21, 130, 888, 6408, 48063, 371020, 2927770, 23510106, 191487656, 1578141446, 13136158810, 110276181794, 932581374945, 7937416827088, 67940240601388, 584459497206176, 5050440687673800, 43818234017497418, 381559984178906078, 3333557306752862686, 29212187993468834734, 256697187873464085446
Offset: 1

Views

Author

Paul D. Hanna, Jul 10 2025

Keywords

Comments

Compare to C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjecture: a(n) is odd iff n = 2^k for k >= 0.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 130*x^5 + 888*x^6 + 6408*x^7 + 48063*x^8 + 371020*x^9 + 2927770*x^10 + 23510106*x^11 + 191487656*x^12 + ...
where A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3).
RELATED SERIES.
A(x^4 + 4*x*A(x)^4) = x^4 + 4*x^5 + 16*x^6 + 88*x^7 + 545*x^8 + 3676*x^9 + 26368*x^10 + 197048*x^11 + 1517156*x^12 + ...
A(x^3 + 3*x*A(x)^3) = x^3 + 3*x^4 + 9*x^5 + 46*x^6 + 270*x^7 + 1755*x^8 + 12325*x^9 + 90774*x^10 + 691398*x^11 + 5401614*x^12 + ...
A(x)^3 / A(x^3 + 3*x*A(x)^3) = 1 + 6*x^2 + 24*x^3 + 180*x^4 + 1272*x^5 + 9468*x^6 + 72528*x^7 + 568890*x^8 + 4544616*x^9 + 36854304*x^10 + 302604984*x^11 + 2510760870*x^12 + ...
which also equals A(x)^4 / A(x^4 + 4*x*A(x)^4).

Links

Crossrefs

Cf. A376226.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^4 + 4*x*Ax^4 ) - Ax*subst(Ax, x, x^3 + 3*x*Ax^3 ), #A+2)); A[n+1]}
for(n=1, 25, print1(a(n), ", "))

Formula

a(n) ~ c * d^n / n^(3/2), where d = 9.35905430294846226886359252511409062627992765028... and c = 0.016106850486841401468225701097845183576331749... - Vaclav Kotesovec, Jul 16 2025

A384830 G.f. satisfies A(x) = A(x^3 - 3*x*A(x)^3) / A(x^2 - 2*x*A(x)^2).

Original entry on oeis.org

1, -1, 4, -21, 110, -616, 3738, -23619, 152470, -1003776, 6726702, -45720504, 314307018, -2181641134, 15269811260, -107651952999, 763745165826, -5448656285938, 39063995033178, -281309141648214, 2033846965665156, -14757571862304042, 107431429198117338, -784411267743868602, 5743068864740600214
Offset: 1

Views

Author

Paul D. Hanna, Jul 11 2025

Keywords

Comments

Compare to C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjectures:
(C.1) a(n) is odd iff n = 2^k for k >= 0.
(C.2) a(n) == 1 (mod 3) iff n = 3^k for k >= 0.
(C.3) a(n) == 2 (mod 3) iff n = A038464(k)/2 for k >= 1, where A038464 lists sums of 2 distinct powers of 3.

Examples

			G.f.: A(x)  = x - x^2 + 4*x^3 - 21*x^4 + 110*x^5 - 616*x^6 + 3738*x^7 - 23619*x^8 + 152470*x^9 - 1003776*x^10 + 6726702*x^11 - 45720504*x^12 + 314307018*x^13 - 2181641134*x^14 + 15269811260*x^15 - 107651952999*x^16 +- ...
where A(x) = A(x^3 - 3*x*A(x)^3) / A(x^2 - 2*x*A(x)^2).
RELATED SERIES.
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 50*x^5 + 278*x^6 - 1620*x^7 + 10029*x^8 - 64262*x^9 + 420054*x^10 - 2793960*x^11 + 18874530*x^12 + ...
A(x)^3 = x^3 - 3*x^4 + 15*x^5 - 88*x^6 + 516*x^7 - 3123*x^8 + 19771*x^9 - 128748*x^10 + 853182*x^11 - 5739306*x^12 + ...
A(x^3 - 3*x*A(x)^3) = x^3 - 3*x^4 + 9*x^5 - 46*x^6 + 270*x^7 - 1575*x^8 + 9517*x^9 - 60228*x^10 + 391950*x^11 - 2595666*x^12 + ...
A(x^2 - 2*x*A(x)^2) = x^2 - 2*x^3 + 3*x^4 - 14*x^5 + 92*x^6 - 528*x^7 + 3027*x^8 - 18674*x^9 + 120414*x^10 - 790332*x^11 + 5264400*x^12 + ...
A(x)^2 / A(x^2 - 2*x*A(x)^2) = 1 + 6*x^2 - 24*x^3 + 120*x^4 - 696*x^5 + 4362*x^6 - 27720*x^7 + 179496*x^8 - 1188324*x^9 + 8004222*x^10 - 54609300*x^11 + 376571358*x^12 + ...
which also equals A(x)^3 / A(x^3 - 3*x*A(x)^3).
SPECIFIC VALUES.
A(t) = 1/9 at t = 0.121516943263807312205895948801335726496880098390997...
A(t) = 1/10 at t = 0.10853522544585482043493483988448061537900985360239...
  where 1/10 = A(t^3 - 3*t/10^3) / A(t^2 - 2*t/10^2).
A(1/8) = 0.11406786932603073004140288621992624859772243547241...
  where A(1/8) = A(1/8^3 - 3/8*A(1/8)^3) / A(1/8^2 - 2/8*A(1/8)^2).
A(1/9) = 0.10221632839303036250437950418981106643715860663579...
A(1/10) = 0.09261382926552257152263444179118404183078004746053...
A(1/11) = 0.08467305103076088485212166668503171895257965884840...
A(1/12) = 0.07799525781645435314575205510854905593664831977089...
A(-1/8) = -0.1721552141574965794714379396217931153561352397408...
A(-1/9) = -0.1381403570271418152616530595087372513703838778026...
A(-1/10) = -0.1188984739221037989978618004289786780166116550840...

Links

Crossrefs

Cf. A038464, A376226, A384270.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 - 3*x*Ax^3 ) - Ax*subst(Ax, x, x^2 - 2*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(a(n), ", "))

Formula

a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 7.79529565596481925683030276174290122793451926009119... and c = 0.0350717712305315691918705178165312896756485548321... - Vaclav Kotesovec, Jul 16 2025
Showing 1-3 of 3 results.

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