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A227372 G.f.: A(x,q) = 1 + x*A(q*x,q) * A(x,q)^2. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1350 (rows 0..20 of triangle, flattened).

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.

EXAMPLE

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; ...

PROG

(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(""))

CROSSREFS

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

Sequence in context: A248669 A103718 A113350 * A164678 A164679 A275183

Adjacent sequences:  A227369 A227370 A227371 * A227373 A227374 A227375

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Jul 09 2013

STATUS

approved

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Last modified February 27 10:15 EST 2020. Contains 332304 sequences. (Running on oeis4.)