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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A227372 G.f.: A(x,q) = 1 + x*A(q*x,q) * A(x,q)^2.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 2, 1, 14, 15, 10, 9, 4, 2, 1, 42, 56, 45, 43, 34, 23, 14, 9, 4, 2, 1, 132, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1, 429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1, 1430, 3003
Offset: 0

Views

Author

Paul D. Hanna, Jul 09 2013

Keywords

Examples

			Triangle begins:
[1];
[1];
[2, 1];
[5, 4, 2, 1];
[14, 15, 10, 9, 4, 2, 1];
[42, 56, 45, 43, 34, 23, 14, 9, 4, 2, 1];
[132, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1];
[429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1];
[1430, 3003, 3564, 3942, 3990, 3921, 3848, 3681, 3242, 2732, 2267, 1838, 1489, 1189, 909, 671, 494, 345, 252, 173, 109, 71, 46, 27, 14, 9, 4, 2, 1]; ...
Explicitly, the polynomials in q begin:
1;
1;
2 + q;
5 + 4*q + 2*q^2 + q^3;
14 + 15*q + 10*q^2 + 9*q^3 + 4*q^4 + 2*q^5 + q^6;
42 + 56*q + 45*q^2 + 43*q^3 + 34*q^4 + 23*q^5 + 14*q^6 + 9*q^7 + 4*q^8 + 2*q^9 + q^10;
132 + 210*q + 196*q^2 + 196*q^3 + 174*q^4 + 156*q^5 + 121*q^6 + 85*q^7 + 59*q^8 + 42*q^9 + 27*q^10 + 14*q^11 + 9*q^12 + 4*q^13 + 2*q^14 + q^15; ...

Links

Crossrefs

Cf. A227373, A227377, A138158, A000108, A001764.

Programs

  • PARI
    {T(n,k)=local(A=1);for(i=1,n,A=1+x*subst(A,x,q*x)*A^2 +x*O(x^n));polcoeff(polcoeff(A,n,x),k,q)}
for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

Formula

T(n,k) = [x^n*q^k] A(x,q) for k=0..n*(n-1)/2, n>=0.
Column 0 is the Catalan numbers (A000108): T(n,0) = C(2*n,n)/(n+1).
Row sums equal A001764: Sum_{k=0..n*(n-1)/2} T(n,k) = C(3*n,n)/(2*n+1).
Antidiagonal sums equal A227373.
Limit of rows, when read in reverse, yields A227377.

A227377 Limit of rows, when read in reverse, of triangle A227372.

Original entry on oeis.org

1, 2, 4, 9, 14, 27, 46, 71, 113, 185, 280, 409, 614, 899, 1325, 1892, 2639, 3717, 5216, 7221, 9990, 13600, 18315, 24705, 33190, 44338, 58998, 78151, 102492, 133963, 174840, 227180, 294463, 380480, 489606, 628157, 801699, 1019864, 1295760, 1641900, 2074523
Offset: 0

Views

Author

Paul D. Hanna, Jul 10 2013

Keywords

Comments

The g.f. of triangle A227372 satisfies: G(x,q) = 1 + x*G(q*x,q)*G(x,q)^2.
What is the generating function for this sequence?

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 59*x^5 + 199*x^6 + 693*x^7 +...

Crossrefs

Cf. A227372, A227373.

Programs

  • PARI
    /* G.f. of A227372: G(x,q) = 1 + x*G(q*x,q)*G(x,q)^2: */
{A227372(n,k)=local(G=1);for(i=1,n,G=1+x*subst(G,x,q*x)*G^2 +x*O(x^n));polcoeff(polcoeff(G,n,x),k,q)}
{a(n)=A227372(n+1, n*(n-1)/2)}
for(n=0,40,print1(a(n),", "))
Showing 1-2 of 2 results.

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