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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. 32
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

See related triangle A138158.

Row sums are the Catalan numbers (A000108), set q=1 in the g.f. to see this.

Antidiagonal sums equal A005169, the number of fountains of n coins.

The maximum in each row of the triangle is A274291. - Torsten Muetze, Nov 28 2018

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

LINKS

Paul D. Hanna and Seiichi Manyama, Table of n, a(n) for n = 0..9919 (rows n=0..39 of triangle, flattened). (first 1351 terms from Paul D. Hanna)

P. Bala, Illustration of triangular decomposition of area beneath Dyck paths

P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454, Section 4

Luca Ferrari, Unimodality and Dyck paths, arXiv:1207.7295 [math.CO], 2012.

FindStat - Combinatorial Statistic Finder, The bounce statistic of a Dyck path, The diagonal inversion statistic of a Dyck path, The area of a Dyck path.

Thomas Prellberg, Area-perimeter generating functions of lattice walks: q-series and their asymptotics, Slides, School of Mathematical Sciences, Queen Mary, University of London, July 1, 2009.

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

FORMULA

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.

G.f. satisfies: A(x,q) = P(x,q)/Q(x,q), where

  P(x,q) = Sum_{n>=0} q^(n^2) * (-x)^n / Product_{k=1..n} (1-q^k),

  Q(x,q) = Sum_{n>=0} q^(n*(n-1)) * (-x)^n / Product_{k=1..n} (1-q^k),

due to Ramanujan's continued fraction identity.

...

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.

Logarithmic derivative of the g.f. A(x,q), wrt x, yields triangle A227532.

From Peter Bala, Jul 11 2019: (Start)

(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.

1/A(q*x,q) is the generating function for the triangle A047998. (End)

EXAMPLE

G.f.: A(x,q) = 1 + x*(1) + x^2*(1 + q) + x^3*(1 + 2*q + q^2 + q^3)

+ x^4*(1 + 3*q + 3*q^2 + 3*q^3 + 2*q^4 + q^5 + q^6)

+ 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)

+ 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) +...

where g.f.:

A(x,q) = Sum_{k=0..n*(n-1)/2, n>=0} T(n,k)*x^n*q^k

satisfies:

A(x,q) = 1 + x*A(q*x,q)*A(x,q).

This triangle of coefficients T(n,k) in A(x,q) begins:

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, 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, 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;

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; ...

MATHEMATICA

T[n_, k_] := Module[{P, Q},

P = Sum[q^(m^2) (-x)^m/Product[1-q^j, {j, 1, m}] + x O[x]^n, {m, 0, n}];

Q = Sum[q^(m(m-1)) (-x)^m/Product[1-q^j, {j, 1, m}] + x O[x]^n, {m, 0, n}];

SeriesCoefficient[P/Q, {x, 0, n}, {q, 0, k}]

];

Table[T[n, k], {n, 0, 10}, {k, 0, n(n-1)/2}] // Flatten (* Jean-François Alcover, Jul 27 2018, from PARI *)

PROG

(PARI) /* From g.f. A(x, q) = 1 + x*A(q*x, q)*A(x, q): */

{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)}

for(n=0, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))

(PARI) /* By Ramanujan's continued fraction identity: */

{T(n, k)=local(P=1, Q=1);

P=sum(m=0, n, q^(m^2)*(-x)^m/prod(k=1, m, 1-q^k)+x*O(x^n));

Q=sum(m=0, n, q^(m*(m-1))*(-x)^m/prod(k=1, m, 1-q^k)+x*O(x^n));

polcoeff(polcoeff(P/Q, n, x), k, q)}

for(n=0, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A138158, A227532, A005169, A000108, A274291, A047998, A224704, A239927.

Sequence in context: A047010 A047100 A124772 * A079415 A126347 A309240

Adjacent sequences:  A227540 A227541 A227542 * A227544 A227545 A227546

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Jul 15 2013

STATUS

approved

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Last modified September 30 05:03 EDT 2020. Contains 337435 sequences. (Running on oeis4.)