A138158 -id:A138158 - cp's OEIS frontend

A005169 Number of fountains of n coins.

1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 135, 234, 406, 704, 1222, 2120, 3679, 6385, 11081, 19232, 33379, 57933, 100550, 174519, 302903, 525734, 912493, 1583775, 2748893, 4771144, 8281088, 14373165, 24946955, 43299485, 75153286, 130440740, 226401112, 392955956, 682038999, 1183789679, 2054659669, 3566196321, 6189714276

Offset: 0

N. J. A. Sloane

A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row.
Also the number of Dyck paths for which the sum of the heights of the vertices that terminate an upstep (i.e., peaks and doublerises) is n. Example: a(4)=3 because we have UDUUDD, UUDDUD and UDUDUDUD. - Emeric Deutsch, Mar 22 2008
Also the number of ordered trees with path length n (follows from previous comment via a standard bijection). - Emeric Deutsch, Mar 22 2008
Probably first studied by Jim Propp (unpublished).
Number of compositions of n with c(1) = 1 and c(i+1) <= c(i) + 1. (Slide each row right 1/2 step relative to the row below, and count the columns.) - Franklin T. Adams-Watters, Nov 24 2009
With the additional requirement for weak unimodality one obtains A001524. - Joerg Arndt, Dec 09 2012

			An example of a fountain with 19 coins:
... O . O O
.. O O O O O O . O
. O O O O O O O O O
From _Peter Bala_, Dec 26 2012: (Start)
F(1/10) = Sum_{n >= 0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(8 + 1/(1 + 1/(8 + 1/(1 + 1/(98 + 1/(1 + 1/(98 + 1/(1 + 1/(998 + 1/(1 + 1/(998 + 1/(1 + ...)))))))))))).
F(-1/10) = Sum_{n >= 0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(9 + 1/(1 + 1/(9 + 1/(99 + 1/(1 + 1/(99 + 1/(999 + 1/(1 + 1/(999 + 1/(9999 + 1/(1 + ...)))))))))))).
(End)

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..4178 (first 501 terms from T. D. Noe)
Peter Bala, Some simple continued fraction expansions
P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 331.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 4.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 2.
M. L. Glasser, V. Privman, N. M. Svrakic, Temperley's triangular lattice compact cluster model: exact solution in terms of the q series. J. Phys. A 20 (1987), no. 18, L1275-L1280.
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kival Ngaokrajang, Illustration for initial terms
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 10.7 (pdf, ps)
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Cf. A001524, A192728, A192729, A192730, A111317, A143951, A285903, A226999 (inverse Euler transform), A291148 (convolution inverse).
First column of A168396. - Franklin T. Adams-Watters, Nov 24 2009
Diagonal of A185646.
Row sums of A047998. Column sums of A138158. - Emeric Deutsch, Mar 22 2008

a005169 0 = 1
a005169 n = a168396 n 1  -- Reinhard Zumkeller, Sep 13 2013; corrected by R. J. Mathar, Sep 16 2013

P[0]:=1: for n to 40 do P[n]:=sort(expand(t*(sum(P[j]*P[n-j-1]*t^(n-j-1),j= 0..n-1)))) end do: F:=sort(sum(P[k],k=0..40)): seq(coeff(F,t,j),j=0..36); # Emeric Deutsch, Mar 22 2008
# second Maple program:
A005169_G:= proc(x,NK); Digits:=250; Q2:=1;
        for k from NK by -1 to 0 do  Q1:=1-x^k/Q2; Q2:=Q1; od;
        Q3:=Q2; S:=1-Q3;
end:
series(A005169_G(x, 20), x, 21); # Sergei N. Gladkovskii, Dec 18 2011

m = 36; p[0] = 1; p[n_] := p[n] = Expand[t*Sum[p[j]*p[n-j-1]*t^(n-j-1), {j, 0, n-1}]]; f[t_] = Sum[p[k], {k, 0, m}]; CoefficientList[Series[f[t], {t, 0, m}], t] (* Jean-François Alcover, Jun 21 2011, after Emeric Deutsch *)
max = 43; Series[1-Fold[Function[1-x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Sep 16 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j], {j, 1, Min[i+1, n]}]];
c[n_] :=  b[n, 0] - b[n-1, 0];
c /@ Range[0, 50] // Accumulate  (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz in A289080 *)

/* using the g.f. from p. L1278 of the Glasser, Privman, Svrakic paper */
N=30;  x='x+O('x^N);
P(k)=sum(n=0,N, (-1)^n*x^(n*(n+1+k))/prod(j=1,n,1-x^j));
G=1+x*P(1)/( (1-x)*P(1)-x^2*P(2) );
Vec(G) /* Joerg Arndt, Feb 10 2011 */

/* As a continued fraction: */
{a(n)=local(A=1+x,CF);CF=1+x;for(k=0,n,CF=1/(1-x^(n-k+1)*CF+x*O(x^n));A=CF);polcoeff(A,n)} /* Paul D. Hanna */

/* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=1+x,P,Q);
P=sum(m=0,sqrtint(n),(-1)^m*x^(m*(m+1))/prod(k=1,m,1-x^k));
Q=sum(m=0,sqrtint(n),(-1)^m*x^(m^2)/prod(k=1,m,1-x^k));
A=P/(Q+x*O(x^n));polcoeff(A,n)}  /* Paul D. Hanna */

A005169(n) = f(n, 1), where f(n, p) = 0 if p > n, 1 if p = n, Sum(1 <= q <= p+1; f(n-p, q)) if p < n. f=A168396.
G.f.: F(t) = Sum_{k>=0} P[k], where P[0]=1, P[n] = t*Sum_{j= 0..n-1} P[j]*P[n-j-1]*t^(n-j-1) for n >= 1. - Emeric Deutsch, Mar 22 2008
G.f.: 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(...)))))) [given on the first page of the Odlyzko/Wilf reference]. - Joerg Arndt, Mar 08 2011
G.f.: 1/G(0), where G(k)= 1 - x^(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
G.f.: A(x) = P(x)/Q(x) where
  P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
  Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity. - Paul D. Hanna, Jul 08 2011
From Peter Bala, Dec 26 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 3, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-2 + 1/(1 + 1/(n-2 + 1/(1 + 1/(n^2-2 + 1/(1 + 1/(n^2-2 + 1/(1 + ...)))))))), while for n >= 2, F(-1/n) has the simple continued fraction expansion 1/(1 + 1/(n-1 + 1/(1 + 1/(n-1 + 1/(n^2-1 + 1/(1 + 1/(n^2-1 + 1/(n^3-1 + 1/(1 + ...))))))))). Examples are given below. Cf. A111317 and A143951.
(End)
a(n) = c * x^(-n) + O((5/3)^n), where c = 0.312363324596741... and x = A347901 = 0.576148769142756... is the lowest root of the equation Q(x) = 0, Q(x) see above (Odlyzko & Wilf 1988). - Vaclav Kotesovec, Jul 18 2013, updated Sep 24 2020
G.f.: G(0), where G(k)= 1 - x^(k+1)/(x^(k+1) - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 06 2013
G.f.: 1 - 1/x + 1/(x*W(0)), where W(k)= 1 - x^(2*k+2)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013

More terms from David W. Wilson, Apr 30 2001

A227543 Triangle defined by g.f. A(x,q) such that: A(x,q) = 1 + xA(qx,q)A(x,q), as read by terms k=0..n(n-1)/2 in rows n>=0.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Jul 15 2013

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
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

			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; ...

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)
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 4.
Peter Bala, Illustration of triangular decomposition of area beneath Dyck paths
Peter 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.
J. Haglund, Catalan Paths and q,t-Enumeration.
Stéphane Ouvry and Alexios P. Polychronakos, Exclusion statistics for particles with a discrete spectrum, arXiv:2105.14042 [cond-mat.stat-mech], 2021.
Hao Pan, A finite field approach to the Carlitz-Riordan q-Catalan numbers, Journal of Algebraic Combinatorics: An International Journal: Volume 56, Issue 4, Dec 2022, pp 1005-1009.
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.

Cf. A129175, A138158, A227532, A005169, A000108, A274291, A047998, A224704, A239927.

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 *)

/* 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(""))

/* 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(""))

P(x, n) =
{
    if ( n<=1, return(1) );
    return( sum( i=0, n-1, P(x, i) * P(x, n-1 -i) * x^((i+1)*(n-1 -i)) ) );
}
for (n=0, 10, print( Vec( P(x, n) ) ) ); \\ Joerg Arndt, Jan 23 2024

\\ faster with memoization:
N=11;
VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
for (n=0, N, print( Vec( P(n) ) ) ); \\ Joerg Arndt, Jan 23 2024

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)
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

A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 1, 0, 6, 0, 6, 0, 1, 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1, 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1, 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1, 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1, 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1

Offset: 0

Joerg Arndt, Mar 29 2014

Triangle A129182 transposed.
Column sums give the Catalan numbers (A000108).
Row sums give A143951.
Sums along falling diagonals give A005169.
T(4n,2n) = A240008(n). - Alois P. Heinz, Mar 30 2014

			Triangle begins:
00:  1;
01:  0, 1;
02:  0, 0, 1;
03:  0, 0, 0, 1;
04:  0, 0, 1, 0, 1;
05:  0, 0, 0, 2, 0, 1;
06:  0, 0, 0, 0, 3, 0, 1;
07:  0, 0, 0, 1, 0, 4, 0, 1;
08:  0, 0, 0, 0, 3, 0, 5, 0, 1;
09:  0, 0, 0, 1, 0, 6, 0, 6, 0, 1;
10:  0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;
11:  0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;
12:  0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;
13:  0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;
14:  0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;
15:  0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;
16:  0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;
17:  0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;
18:  0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;
19:  0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;
20:  0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;
...
Column k=4 corresponds to the following 14 paths (dots denote zeros):
#:         path              area   steps (Dyck word)
01:  [ . 1 . 1 . 1 . 1 . ]     4     + - + - + - + -
02:  [ . 1 . 1 . 1 2 1 . ]     6     + - + - + + - -
03:  [ . 1 . 1 2 1 . 1 . ]     6     + - + + - - + -
04:  [ . 1 . 1 2 1 2 1 . ]     8     + - + + - + - -
05:  [ . 1 . 1 2 3 2 1 . ]    10     + - + + + - - -
06:  [ . 1 2 1 . 1 . 1 . ]     6     + + - - + - + -
07:  [ . 1 2 1 . 1 2 1 . ]     8     + + - - + + - -
08:  [ . 1 2 1 2 1 . 1 . ]     8     + + - + - - + -
09:  [ . 1 2 1 2 1 2 1 . ]    10     + + - + - + - -
10:  [ . 1 2 1 2 3 2 1 . ]    12     + + - + + - - -
11:  [ . 1 2 3 2 1 . 1 . ]    10     + + + - - - + -
12:  [ . 1 2 3 2 1 2 1 . ]    12     + + + - - + - -
13:  [ . 1 2 3 2 3 2 1 . ]    14     + + + - + - - -
14:  [ . 1 2 3 4 3 2 1 . ]    16     + + + + - - - -
There are no paths with weight < 4, one with weight 4, none with weight 5, 3 with weight 6, etc., therefore column k=4 is
[0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, ...].
Row n=8 is [0, 0, 0, 0, 3, 0, 5, 0, 1], the corresponding paths of weight=8 are:
Semilength 4:
  [ . 1 . 1 2 1 2 1 . ]
  [ . 1 2 1 . 1 2 1 . ]
  [ . 1 2 1 2 1 . 1 . ]
Semilength 6:
  [ . 1 . 1 . 1 . 1 . 1 2 1 . ]
  [ . 1 . 1 . 1 . 1 2 1 . 1 . ]
  [ . 1 . 1 . 1 2 1 . 1 . 1 . ]
  [ . 1 . 1 2 1 . 1 . 1 . 1 . ]
  [ . 1 2 1 . 1 . 1 . 1 . 1 . ]
Semilength 8:
  [ . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . ]

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Sequences obtained by particular choices for x and y in the g.f. F(x,y) are: A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(sqrt(x), sqrt(x))),  A227310 (1+x*F(x, x^2), also 2-1/F(x, 1)), A239928 (F(x^2, x)), A052709 (x*F(1,x+x^2)), A125305 (F(1, x+x^3)), A002212 (F(1, x/(1-x))).
Cf. A047998, A138158, A227543.
Cf. A129181.

b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0, 0, `if`(x=0, `if`(k=0, 1, 0),
       b(x-1, y-1, k-y+1/2)+ b(x-1, y+1, k-y-1/2)))
    end:
T:= (n, k)-> b(2*k, 0, n):
seq(seq(T(n, k), k=0..n), n=0..20);  # Alois P. Heinz, Mar 29 2014

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0, 0, If[x == 0, If[k == 0, 1, 0], b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]]; T[n_, k_] := b[2*k, 0, n]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }
print_triangle(V)= { my( N=#V ); for(n=1, N, print( rvec( V[n]) ) ); }
N=20; x='x+O('x^N);
F(x,y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
v= Vec( F(x,y) );
print_triangle(v)

G.f.: F(x,y) satisfies F(x,y) = 1 / (1 - x*y * F(x, x^2*y) ).

G.f.: 1/(1 - y*x/(1 - y*x^3/(1 - y*x^5/(1 - y*x^7/(1 - y*x^9/( ... )))))).

A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 2, 2, -4, 0, 0, 2, 4, -8, 0, 0, 4, 8, -14, 0, 0, 8, 14, -24, 0, 0, 12, 22, -40, 0, 0, 20, 36, -64, 0, 0, 32, 56, -98, 0, 0, 48, 84, -148, 0, 0, 72, 126, -220, 0, 0, 106, 184, -320, 0, 0, 152, 264, -460, 0, 0, 216, 376, -652, 0, 0, 306, 528, -912, 0, 0, 424, 732, -1264, 0, 0, 584, 1008

Offset: 0

Michael Somos, Mar 23 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Denoted by t in Andrews and Berndt 2005. - Michael Somos, Apr 25 2016

			G.f. = 1 - 2*q + 2*q^4 + 2*q^5 - 4*q^6 + 2*q^9 + 4*q^10 - 8*q^11 + 4*q^14 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A116494, A138158, A138526.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5], {q, 0, n}]; (* Michael Somos, Sep 13 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A), n))};

Expansion of (eta(q) / eta(q^5))^2 * eta(q^10) / eta(q^2) in powers of q.
Euler transform of period 10 sequence [ -2, -1, -2, -1, 0, -1, -2, -1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u^2)^2 - u^2 * (1 - v^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (1 - u^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u + v)^2 * w^2 - u * v * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 * u3 - u1 * u6)^2 - u1 * u3 * (u6^2 - u2^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116494.
G.f.: Product_{k>0} P(10, x^k) / P(5, x^k) where P(n, x) is the n-th cyclotomic polynomial.
a(5*n + 2) = a(5*n + 3) = 0.
Convolution inverse is A138526. Convolution square is A138518.

A227372 G.f.: A(x,q) = 1 + xA(qx,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

Paul D. Hanna, Jul 09 2013

			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; ...

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

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

{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(""))

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.

A274291 The width of the lattice of Dyck paths of length 2n ordered by the relation that one Dyck path lies above another one.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 17, 44, 118, 338, 1003, 3039, 9466, 30009, 96757, 316429, 1047683, 3511473, 11876457, 40537388, 139490014, 483393651, 1686007017, 5917253784, 20879801881, 74038098051, 263793988890, 943928231920, 3390975927021, 12227214763162, 44242758258306

Offset: 0

N. J. A. Sloane, Jun 17 2016

nonn

Previous name was: The width of the lattice E_n defined in the paper by Boldi and Vigna, that is, the cardinality of a maximal antichain.

a(n) is the maximum entry in row n of the triangle T(n,k) defined in A138158, or equivalently, the maximum entry in row n of the triangle T(n,k) defined in A227543. All level sizes of the lattice are given by A138158 and A227543. - Torsten Muetze, Nov 28 2018

			For n=4 there are 14 Dyck paths, and 1,3,3,3,2,1,1 of them have area 0,1,2,3,4,5,6, respectively, where the area is normalized to the range 0,...,n(n-1)/2. These Dyck paths are UDUDUDUD (area=0), UUDDUDUD, UDUUDDUD, UDUDUUDD (area=1), UUDUDDUD, UDUUDUDD, UUDDUUDD (area=2), UUUDDDUD, UUDUDUDD, UDUUUDDD (area=3), UUUDDUDD, UUDUUDDD (area=4), UUUDUDDD (area=5), UUUUDDDD (area=6). The maximum among the numbers 1,3,3,3,2,1,1 is 3, so a(4)=3.

Winston, Kenneth J., and Daniel J. Kleitman. "On the asymptotic number of tournament score sequences." Journal of Combinatorial Theory, Series A 35.2 (1983): 208-230. See Table 1.

Torsten Muetze, Table of n, a(n) for n = 0..300
Paolo Boldi and Sebastiano Vigna, On the Lattice of Antichains of Finite Intervals, Order (2016), 1-25.
Paolo Boldi, Sebastiano Vigna, On the lattice of antichains of finite intervals, arXiv preprint arXiv:1510.03675 [math.CO], 2015-2016.

Cf. A138158, A227543.

a(0)=1 inserted by Sebastiano Vigna, Dec 20 2017

New name and more terms from Torsten Muetze, Nov 28 2018

cp's OEIS Frontend

A005169 Number of fountains of n coins.

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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.

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A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

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A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.

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

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A274291 The width of the lattice of Dyck paths of length 2n ordered by the relation that one Dyck path lies above another one.

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A227543 Triangle defined by g.f. A(x,q) such that: A(x,q) = 1 + xA(qx,q)A(x,q), as read by terms k=0..n(n-1)/2 in rows n>=0.

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