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.
%I A227543 #96 Apr 14 2025 09:01:59
%S A227543 1,1,1,1,1,2,1,1,1,3,3,3,2,1,1,1,4,6,7,7,5,5,3,2,1,1,1,5,10,14,17,16,
%T A227543 16,14,11,9,7,5,3,2,1,1,1,6,15,25,35,40,43,44,40,37,32,28,22,18,13,11,
%U A227543 7,5,3,2,1,1,1,7,21,41,65,86,102,115,118,118,113,106,96,85,73,63,53,42,34,26,20,15,11,7,5,3,2,1,1
%N A227543 Triangle defined by g.f. A(x,q) such that: A(x,q) = 1 + x*A(q*x,q)*A(x,q), as read by terms k=0..n*(n-1)/2 in rows n>=0.
%C A227543 See related triangle A138158.
%C A227543 Row sums are the Catalan numbers (A000108), set q=1 in the g.f. to see this.
%C A227543 Antidiagonal sums equal A005169, the number of fountains of n coins.
%C A227543 The maximum in each row of the triangle is A274291. - _Torsten Muetze_, Nov 28 2018
%C A227543 The area between a Dyck path and the x-axis may be decomposed into unit area triangles of two types - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, y+1) and (x+1, y+1). The table entry T(n,k) equals the number of Dyck paths of semilength n containing k down triangles. See the illustration in the Links section. Cf. A239927. - _Peter Bala_, Jul 11 2019
%C A227543 The row polynomials of this table are a q-analog of the Catalan numbers due to Carlitz and Riordan. For MacMahon's q-analog of the Catalan numbers see A129175. - _Peter Bala_, Feb 28 2023
%H A227543 Paul D. Hanna and Seiichi Manyama, <a href="/A227543/b227543.txt">Table of n, a(n) for n = 0..9919 (rows n=0..39 of triangle, flattened).</a> (first 1351 terms from Paul D. Hanna)
%H A227543 M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, <a href="https://doi.org/10.1007/s13370-025-01282-0">Subdiagonal and superdiagonal partitions</a>, Afr. Mat. 36, 77 (2025). See p. 4.
%H A227543 Peter Bala, <a href="/A227543/a227543.pdf">Illustration of triangular decomposition of area beneath Dyck paths</a>
%H A227543 Peter Bala, <a href="/A224704/a224704.pdf">The area beneath small Schröder paths: Notes on A224704, A326453 and A326454</a>, Section 4
%H A227543 Luca Ferrari, <a href="http://arxiv.org/abs/1207.7295">Unimodality and Dyck paths</a>, arXiv:1207.7295 [math.CO], 2012.
%H A227543 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000005">The bounce statistic of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000006">The diagonal inversion statistic of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000012">The area of a Dyck path</a>.
%H A227543 J. Haglund, <a href="https://www2.math.upenn.edu/~jhaglund/books/qtcat.pdf">Catalan Paths and q,t-Enumeration</a>.
%H A227543 Stéphane Ouvry and Alexios P. Polychronakos, <a href="https://arxiv.org/abs/2105.14042">Exclusion statistics for particles with a discrete spectrum</a>, arXiv:2105.14042 [cond-mat.stat-mech], 2021.
%H A227543 Hao Pan, <a href="https://doi.org/10.1007/s10801-022-01141-2">A finite field approach to the Carlitz-Riordan q-Catalan numbers</a>, Journal of Algebraic Combinatorics: An International Journal: Volume 56, Issue 4, Dec 2022, pp 1005-1009.
%H A227543 Thomas Prellberg, <a href="http://www.maths.qmul.ac.uk/~tp/talks/asymptotics.pdf"> Area-perimeter generating functions of lattice walks: q-series and their asymptotics</a>, Slides, School of Mathematical Sciences, Queen Mary, University of London, July 1, 2009.
%H A227543 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>.
%F A227543 G.f.: A(x,q) = 1/(1 - x/(1 - q*x/(1 - q^2*x/(1 - q^3*x/(1 - q^4*x/(1 -...)))))), a continued fraction.
%F A227543 G.f. satisfies: A(x,q) = P(x,q)/Q(x,q), where
%F A227543 P(x,q) = Sum_{n>=0} q^(n^2) * (-x)^n / Product_{k=1..n} (1-q^k),
%F A227543 Q(x,q) = Sum_{n>=0} q^(n*(n-1)) * (-x)^n / Product_{k=1..n} (1-q^k),
%F A227543 due to Ramanujan's continued fraction identity.
%F A227543 ...
%F A227543 Sum_{k=0..n*(n-1)/2} T(n,k)*k = 2^(2*n-1) - C(2*n+1,n) + C(2*n-1,n-1) = A006419(n-1) for n>=1.
%F A227543 Logarithmic derivative of the g.f. A(x,q), wrt x, yields triangle A227532.
%F A227543 From _Peter Bala_, Jul 11 2019: (Start)
%F A227543 (n+1)th row polynomial R(n+1,q) = Sum_{k = 0..n} q^k*R(k,x)*R(n-k,q), with R(0,q) = 1.
%F A227543 1/A(q*x,q) is the generating function for the triangle A047998. (End)
%F A227543 Conjecture: b(n) = P(n, n) where b(n) is an integer sequence with g.f. B(x) = 1/(1 - f(0)*x/(1 - f(1)*x/(1 - f(2)*x/(1 - f(3)*x/(1 - f(4)*x/(1 -...)))))), P(n, k) = P(n-1, k) + f(n-k)*P(n, k-1) for 0 < k <= n with P(n, k) = 0 for k > n, P(n, 0) = 1 for n >= 0 and where f(n) is an arbitrary function. In fact for this sequence we have f(n) = q^n. - _Mikhail Kurkov_, Sep 26 2024
%e A227543 G.f.: A(x,q) = 1 + x*(1) + x^2*(1 + q) + x^3*(1 + 2*q + q^2 + q^3)
%e A227543 + x^4*(1 + 3*q + 3*q^2 + 3*q^3 + 2*q^4 + q^5 + q^6)
%e A227543 + x^5*(1 + 4*q + 6*q^2 + 7*q^3 + 7*q^4 + 5*q^5 + 5*q^6 + 3*q^7 + 2*q^8 + q^9 + q^10)
%e A227543 + x^6*(1 + 5*q + 10*q^2 + 14*q^3 + 17*q^4 + 16*q^5 + 16*q^6 + 14*q^7 + 11*q^8 + 9*q^9 + 7*q^10 + 5*q^11 + 3*q^12 + 2*q^13 + q^14 + q^15) +...
%e A227543 where g.f.A(x,q) = Sum_{k=0..n*(n-1)/2, n>=0} T(n,k)*x^n*q^k
%e A227543 satisfies A(x,q) = 1 + x*A(q*x,q)*A(x,q).
%e A227543 This triangle of coefficients T(n,k) in A(x,q) begins:
%e A227543 1;
%e A227543 1;
%e A227543 1, 1;
%e A227543 1, 2, 1, 1;
%e A227543 1, 3, 3, 3, 2, 1, 1;
%e A227543 1, 4, 6, 7, 7, 5, 5, 3, 2, 1, 1;
%e A227543 1, 5, 10, 14, 17, 16, 16, 14, 11, 9, 7, 5, 3, 2, 1, 1;
%e A227543 1, 6, 15, 25, 35, 40, 43, 44, 40, 37, 32, 28, 22, 18, 13, 11, 7, 5, 3, 2, 1, 1;
%e A227543 1, 7, 21, 41, 65, 86, 102, 115, 118, 118, 113, 106, 96, 85, 73, 63, 53, 42, 34, 26, 20, 15, 11, 7, 5, 3, 2, 1, 1;
%e A227543 1, 8, 28, 63, 112, 167, 219, 268, 303, 326, 338, 338, 331, 314, 293, 268, 245, 215, 190, 162, 139, 116, 97, 77, 63, 48, 38, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1; ...
%t A227543 T[n_, k_] := Module[{P, Q},
%t A227543 P = Sum[q^(m^2) (-x)^m/Product[1-q^j, {j, 1, m}] + x O[x]^n, {m, 0, n}];
%t A227543 Q = Sum[q^(m(m-1)) (-x)^m/Product[1-q^j, {j, 1, m}] + x O[x]^n, {m, 0, n}];
%t A227543 SeriesCoefficient[P/Q, {x, 0, n}, {q, 0, k}]
%t A227543 ];
%t A227543 Table[T[n, k], {n, 0, 10}, {k, 0, n(n-1)/2}] // Flatten (* _Jean-François Alcover_, Jul 27 2018, from PARI *)
%o A227543 (PARI) /* From g.f. A(x,q) = 1 + x*A(q*x,q)*A(x,q): */
%o A227543 {T(n, k)=local(A=1); for(i=1, n, A=1+x*subst(A, x, q*x)*A +x*O(x^n)); polcoeff(polcoeff(A, n, x), k, q)}
%o A227543 for(n=0, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))
%o A227543 (PARI) /* By Ramanujan's continued fraction identity: */
%o A227543 {T(n,k)=local(P=1,Q=1);
%o A227543 P=sum(m=0,n,q^(m^2)*(-x)^m/prod(k=1,m,1-q^k)+x*O(x^n));
%o A227543 Q=sum(m=0,n,q^(m*(m-1))*(-x)^m/prod(k=1,m,1-q^k)+x*O(x^n));
%o A227543 polcoeff(polcoeff(P/Q,n,x),k,q)}
%o A227543 for(n=0, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))
%o A227543 (PARI)
%o A227543 P(x, n) =
%o A227543 {
%o A227543 if ( n<=1, return(1) );
%o A227543 return( sum( i=0, n-1, P(x, i) * P(x, n-1 -i) * x^((i+1)*(n-1 -i)) ) );
%o A227543 }
%o A227543 for (n=0, 10, print( Vec( P(x, n) ) ) ); \\ _Joerg Arndt_, Jan 23 2024
%o A227543 (PARI) \\ faster with memoization:
%o A227543 N=11;
%o A227543 VP=vector(N+1); VP[1] =VP[2] = 1; \\ one-based; memoization
%o A227543 P(n) = VP[n+1];
%o A227543 for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
%o A227543 for (n=0, N, print( Vec( P(n) ) ) ); \\ _Joerg Arndt_, Jan 23 2024
%Y A227543 Cf. A129175, A138158, A227532, A005169, A000108, A274291, A047998, A224704, A239927.
%K A227543 nonn,tabf
%O A227543 0,6
%A A227543 _Paul D. Hanna_, Jul 15 2013