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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 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.)