A005169 -id:A005169 - cp's OEIS frontend

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.

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

A023361 Number of compositions of n into positive triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 11, 16, 25, 40, 61, 94, 147, 227, 351, 546, 846, 1309, 2030, 3147, 4876, 7558, 11715, 18154, 28136, 43609, 67586, 104748, 162346, 251610, 389958, 604381, 936699, 1451743, 2249991, 3487153, 5404570, 8376292, 12982016, 20120202, 31183350

Offset: 0

David W. Wilson, Jun 14 1998

nonn

Number of compositions [c(1), c(2), c(3), ...] of n such that either c(k) = c(k-1) + 1 or c(k) = 1; see example. Same as fountains of coins (A005169) where each valley is at the lowest level. - Joerg Arndt, Mar 25 2014

			From _Joerg Arndt_, Mar 25 2014: (Start)
There are a(9) = 25 compositions of 9 such that either c(k) = c(k-1) + 1 or c(k) = 1:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 1 1 1 ]
06:  [ 1 1 1 1 2 1 2 ]
07:  [ 1 1 1 1 2 3 ]
08:  [ 1 1 1 2 1 1 1 1 ]
09:  [ 1 1 1 2 1 1 2 ]
10:  [ 1 1 1 2 1 2 1 ]
11:  [ 1 1 1 2 3 1 ]
12:  [ 1 1 2 1 1 1 1 1 ]
13:  [ 1 1 2 1 1 1 2 ]
14:  [ 1 1 2 1 1 2 1 ]
15:  [ 1 1 2 1 2 1 1 ]
16:  [ 1 1 2 3 1 1 ]
17:  [ 1 2 1 1 1 1 1 1 ]
18:  [ 1 2 1 1 1 1 2 ]
19:  [ 1 2 1 1 1 2 1 ]
20:  [ 1 2 1 1 2 1 1 ]
21:  [ 1 2 1 2 1 1 1 ]
22:  [ 1 2 1 2 1 2 ]
23:  [ 1 2 1 2 3 ]
24:  [ 1 2 3 1 1 1 ]
25:  [ 1 2 3 1 2 ]
The last few, together with the corresponding fountains of coins are:
.  20:  [ 1 2 1 1 2 1 1 ]
.
.     O     O
.    O O O O O O O
.
.
.  21:  [ 1 2 1 2 1 1 1 ]
.
.     O   O
.    O O O O O O O
.
.
.  22:  [ 1 2 1 2 1 2 ]
.
.     O   O   O
.    O O O O O O
.
.
.  23:  [ 1 2 1 2 3 ]
.
.           O
.      O   O O
.     O O O O O
.
.
.  24:  [ 1 2 3 1 1 1 ]
.
.       O
.      O O
.     O O O O O O
.
.
.  25:  [ 1 2 3 1 2 ]
.
.       O
.      O O   O
.     O O O O O
(End)
Applying recursion formula: 40 = a(10) = a(9) + a(7) + a(4) + a(0) = 25 + 11 + 3 + 1. - _Gregory L. Simay_, Jun 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..5256 (terms n = 0..500 from T. D. Noe)
N. Robbins, On compositions whose parts are polygonal numbers, Annales Univ. Sci. Budapest., Sect. Comp. 43 (2014) 239-243. See p. 242.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A106332.

a:= proc(n) option remember; `if`(n=0, 1,
      add(`if`(issqr(8*j+1), a(n-j), 0), j=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017

(1/(2 - QPochhammer[x^2]/QPochhammer[x, x^2]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 23 2016 *)
a[n_] := a[n] = If[n == 0, 1, Sum[ If[ IntegerQ[ Sqrt[8j+1]], a[n-j], 0], {j, 1, n}]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

N=66;  x='x+O('x^N);
Vec( 1/( 1 - sum(k=1,1+sqrtint(2*N), x^binomial(k+1,2) ) ) )
/* Joerg Arndt, Sep 30 2012 */

G.f. : 1 / (1 - Sum_{k>=1} x^(k*(k+1)/2) ).
a(n) ~ c * d^n, where d = 1/A106332 = 1.5498524695188884304192160776463163555... is the root of the equation d^(1/8) * EllipticTheta(2, 0, 1/sqrt(d)) = 4 and c = 0.492059962414480455851222791075288170662444559041717451009563731799... - Vaclav Kotesovec, May 01 2014, updated Feb 17 2017
a(n) = a(n-1) + a(n-3) + a(n-6) + a(n-10) + ...  Gregory L. Simay, Jun 09 2016
G.f.: 1/(2 - (x^2;x^2)inf/(x;x^2)_inf), where (a;q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 23 2016
G.f.: 1/(2 - theta_2(sqrt(q))/(2*q^(1/8))), where theta_2() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A006958 Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused).

Original entry on oeis.org

1, 2, 4, 9, 20, 46, 105, 242, 557, 1285, 2964, 6842, 15793, 36463, 84187, 194388, 448847, 1036426, 2393208, 5526198, 12760671, 29466050, 68041019, 157115917, 362802072, 837759792, 1934502740, 4467033943, 10314998977, 23818760154, 55000815222, 127004500762

Offset: 1

N. J. A. Sloane, Simon Plouffe

Same as: number of skew Ferrers diagrams. - Joerg Arndt, Mar 18 2014

A coin fountain is an arrangement of coins in numbered rows such that the bottom row (row 0) contains contiguous coins and such that each coin in a higher row touches exactly two coins in the next lower row. See A005169. a(n) equals the number of coin fountains with exactly n coins in the even-numbered rows of the fountain. See the illustration in the Links section. See A161492 for a refinement of this sequence. - Peter Bala, Jul 20 2019

			G.f. may be expressed by the continued fraction: 1/(1-x/(1-x/(1-x^2/(1-x^2/(1-x^3/(1-x^3/(1-x^4/(1-...)))))))) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 105*x^7 + ...
From _Michael B. Porter_, Sep 21 2016; corrected by _Riccardo Moschetti_, Aug 11 2017: (Start)
Here are the nine parallelogram polyominoes with 4 cells, i.e., polygons convex according to the -45-degree direction, according to "Polya Festoons" of P. Flajolet:
                          _      _  _
             _  _     _ /_ /   /_ /_ /
         _ /_ /_ /  /_ /_ /   /_ /      _  _  _  _
       /_ /_ /     /_ /      /_ /     /_ /_ /_ /_ /
                     _
              _    /_ /    _  _  _            _  _
            /_ /  /_ /   /_ /_ /_ /    _    /_ /_ /
         _ /_ /  /_ /   /_ /    _  _ /_ /  /_ /_ /
       /_ /_ /  /_ /          /_ /_ /_ /
(End)

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.19, p. 380.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..2752 (terms 1..1000 from Robert Israel)
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Peter Bala, Illustration for the initial terms of the sequence.
Peter Bala, Fountains of coins and skew Ferrers diagrams.
E. A. Bender, Convex n-ominoes, Discrete Math., 8 (1974), 219-226.
M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics, vol. 112 (1993), no. 1-3, pp. 65-79.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
P. Flajolet, Email to N. J. A. Sloane & S. Plouffe, Aug. 1991.
P. Flajolet, Polya Festoons, INRIA Research Report, No 1507, September 1991. 6pp.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 661.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 4-5.
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.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
D. A. Klarner and R. L. Rivest, Asymptotic bounds for the number of convex n-ominoes, Discrete Math., 8 (1974), 31-40.

Cf. A075125, A276994, A275761, A275762, A161492.

N:= 100: # to get a(1) to a(N)
M:= ceil(sqrt(N+1)):
C:= 1:
for j from M to 1 by -1 do C:= 1/(1-x^j/(1-x^j*C)) od:
S:= series(C,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Sep 20 2016

NN = 100; (* to get a(1) to a(NN) *) M = Ceiling[Sqrt[NN+1]]; c = 1; For[j = M, j >= 1, j--, c = 1/(1-x^j/(1-x^j*c))]; c = Series[c, {x, 0, NN+1}]; CoefficientList[c, x][[2 ;; NN+1]] (* Jean-François Alcover, Sep 27 2016, adapted from Robert Israel's Maple code *)
nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1] - Floor[Range[nmax + 1]/2])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

{a(n)=local(CF=1+x*O(x^n),m); for(k=0,n\2,m=n\2-k+1;CF=(1-x^((m+1)\2)/CF));polcoeff(1/CF,n)} \\ Paul D. Hanna, May 14 2005

/* From the Delest/Fedou reference: */
N=44;  q='q+O('q^N);  t=1;
qn(n) = prod(k=1, n, 1-q^k );
nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
Vec(nm/dn)  \\ Joerg Arndt, Mar 18 2014

G.f.: 1+A(x) = 1/(1-x/(1-x/(1-x^2/(1-x^2/(1-x^3/(1-x^3/(1-...))))))) (continued fraction). - Paul D. Hanna, May 14 2005
The continued fraction given by P. Flajolet, "Polya Festoons", is derived from a q-expansion, C(x, y;q), where C(1, 1;q) = q/(1-2*q-q^3/(1-2*q^2-q^5/(1-2*q^3-q^7/(1-2*q^4-q^9/(1-...))))) = q + 2*q^2 + 4*q^3 + 9*q^4 + 20*q^5 + 46*q^6 + 105*q^7 + ... - Paul D. Hanna, May 14 2005
G.f.: 1/x/G(0) -1/x, where G(k)= 1 - x^(k+1)/(1 - x^(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x^(k+1)/( x^(k+1) - 1/(1 - x^(k+1)/( x^(k+1) - 1/W(k+1) ))); R=1 (continued fraction). - Sergei N. Gladkovskii, Aug 27 2013
a(n) ~ c * d^n, where d = A276994 = 2.309138593330494731098720305017212531911814472581628401694402900284456440748..., c = 0.29745350581112195107675842441785013227507248969090226252518932405713367... . - Vaclav Kotesovec, Sep 21 2016
From Peter Bala, Jul 21 2019: (Start)
O.g.f. as a ratio of q-series: 1 + A(q) = N(q)/D(q) = 1 + q + 2*q^2 + 4*q^3 + ..., where N(q) = Sum_{n >= 0} (-1)^n*q^((n^2 + 3*n)/2)/Product_{k = 1..n} (1 - q^k)^2 and D(q) = Sum_{n >= 0} (-1)^n*q^((n^2 + n)/2)/Product_{k = 1..n} (1 - q^k)^2.
The constant d = 2.30913... in the above asymptotic formula is a zero of D(q) (as is 1/d).
Continued fraction representations for the o.g.f.:
1 + A(q) = 1/(1 - q/(1 - q/(1 + q*(1 - q) - q/(1 + q*(1 - q^2) - q/(1 + q*(1 - q^3) - (...) ))))).
1 + A(q) = 1/(1 - q - q^2/(1 - q*(1 + q) - q^4/(1 - q^2*(1 + q) - q^6/(1 - q^3(1 + q) - q^8/( (...) ))))).
1 + A(q) = 1/(1 - q - q^2/(1 - q^2 - q/(1 - q^3 - q^5/(1 - q^4 - q^2/(1 - q^5 - q^8/(1 - q^6 - q^3/(1 - q^7 - q^11/(1 - q^8 - (...) )))))))). (End)

More terms from Paul D. Hanna, May 14 2005

Definition modified by Don Knuth, Sep 20 2016

A008795 Molien series for 3-dimensional representation of dihedral group D_6 of order 6.

Original entry on oeis.org

1, 0, 3, 1, 6, 3, 10, 6, 15, 10, 21, 15, 28, 21, 36, 28, 45, 36, 55, 45, 66, 55, 78, 66, 91, 78, 105, 91, 120, 105, 136, 120, 153, 136, 171, 153, 190, 171, 210, 190, 231, 210, 253, 231, 276, 253, 300, 276, 325, 300, 351, 325, 378, 351, 406, 378, 435, 406, 465, 435, 496, 465, 528, 496

Offset: 0

N. J. A. Sloane

a(n-3) is the number of ordered triples of positive integers which are the side lengths of a nondegenerate triangle of perimeter n. - Rob Pratt, Jul 12 2004
a(n) is the number of ways to distribute n identical objects into 3 distinguishable bins so that no bin contains an absolute majority of objects. - Geoffrey Critzer, Mar 17 2010
From Omar E. Pol, Feb 05 2012: (Start)
Also terms of A000217 and A000217-shifted interleaved.
Also 0 together with this sequence give the first row of the square array A194801. (End)
a(n) is the number of coins left after packing 3-curves coins patterns into fountain of coins base n. Refer to A005169: "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". See illustration in links. - Kival Ngaokrajang, Oct 12 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Kival Ngaokrajang, Illustration of initial terms
Ira Rosenholtz, Problem 1584, Mathematics Magazine, Vol. 72 (1999), p. 408.
Index entries for sequences related to groups
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A005044.

First differences of A053307.

a := [1,0,3,1,6];; for n in [6..70] do a[n] := a[n-1] + 2*a[n-2] -2*a[n-3] -a[n-4] +a[n-5]; od; a; # Muniru A Asiru, Feb 01 2018

[(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16: n in [0..70] ]; // Vincenzo Librandi, Aug 21 2011

a:= n-> binomial(n/2+2-3*irem(n, 2)/2, 2):
seq(a(n), n=0..70); # Muniru A Asiru, Feb 01 2018

Table[If[EvenQ[n], Binomial[n/2+2, 2], Binomial[(n+1)/2, 2]], {n, 0, 70}]
CoefficientList[Series[(1+x^3)/(1-x^2)^3, {x, 0, 70}], x] (* Robert G. Wilson v, Feb 05 2012 *)
a[ n_]:= Binomial[ Quotient[n, 2] + 2 - Mod[n, 2], 2]; (* Michael Somos, Feb 01 2018 *)
a[ n_]:= With[ {m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ (1 - x + x^2) / ((1 - x)^3 (1 + x)^2), {x, 0, m}]]; (* Michael Somos, Feb 01 2018 *)
LinearRecurrence[{1,2,-2,-1,1}, {1,0,3,1,6}, 70] (* Robert G. Wilson v, Feb 01 2018 *)

a(n)=(2*n^2+6*n+7)/16+3*(2*n+3)*(-1)^n/16 \\ Charles R Greathouse IV, Oct 22 2015

{a(n) = binomial(n\2 + 2 - n%2, 2)}; /* Michael Somos, Feb 01 2018 */

[(2*n^2 +6*n +7 +3*(2*n+3)*(-1)^n)/16 for n in (0..70)] # G. C. Greubel, Sep 11 2019

The signed version with g.f. (1-x^3)/(1-x^2)^3 is the inverse binomial transform of A084861. - Paul Barry, Jun 12 2003
a(n) = binomial(n/2+2, 2) for n even, binomial((n+1)/2, 2) for n odd. - Rob Pratt, Jul 12 2004
From Paul Barry, Jul 29 2004: (Start)
a(n-2) interleaves n(n+1)/2 and n(n-1)/2.
G.f.: (1-x+x^2)/((1+x)^2*(1-x)^3).
a(n) = (2*n^2 + 6*n + 7 + 3*(2*n+3)*(-1)^n)/16. (End)
a(n) = n*(n+1)/2, n = +- 1, +- 2... - Omar E. Pol, Feb 05 2012
From Michael Somos, Feb 01 2018: (Start)
Euler transform of length 6 sequence [0, 3, 1, 0, 0, -1].
G.f.: (1 + x^3) / (1 - x^2)^3.
a(n) = a(-3-n) for all in Z. (End)

Definition clarified by N. J. A. Sloane, Feb 02 2018

A224704 Number of stacks of n triangles, pointing upwards or downwards depending on row parity.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 13, 24, 45, 84, 156, 291, 543, 1013, 1889, 3524, 6575, 12266, 22883, 42691, 79647, 148593, 277221, 517197, 964911, 1800189, 3358526, 6265846, 11689902, 21809313, 40688632, 75910917, 141623529, 264220545, 492944193, 919663462, 1715774125

Offset: 0

Paul Tek, Apr 16 2013

nonn

A stack is formed by starting with a row of triangles pointing upwards, and then additional triangles pointing downwards can be inserted between adjacent triangles pointing upwards, and additional triangles pointing upwards can be put on top of triangles pointing downwards.
Number of compositions of n with a(1) = 1 and a(i+1) <= a(i) + 1 + mod(a(i),2).
From Peter Bala, Jul 12 2019: (Start)
These triangle stacks may be defined using Schröder paths. A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.
The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles of two types - up triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..3694 (terms 0..500 from Paul Tek)
P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454
Paul Tek, Illustration of the first terms

Column 1 of A316354 (except a(0)).

Cf. A005169, A001524.

a[ n_] := SeriesCoefficient[ 1/(1 + ContinuedFractionK[ - q^(2 k - 1), 1 - q^(2 k), {k, Ceiling @ Sqrt[n]}]), {q, 0, n}]; (* Michael Somos, Jul 14 2019 *)

use bigint;
my $max = 100;
my @d = ( [1] );
foreach my $n (0..$max) {
    my $a = 0;
    foreach my $h (0..$#{$d[$n]}) {
        $a += $d[$n][$h];
        my $maxh = ($h % 2) ? ($h+2) : ($h+1);
        foreach my $newh (1..$maxh) {
            $d[$n+$newh][$newh] += $d[$n][$h];
        }
    }
    print "$a,";
}

From Peter Bala, Jul 03 2019: (Start)
O.g.f. as a continued fraction: A(q) = 1/(1 - q/(1 - q^2 - q^3/(1 - q^4 - q^5/(1 - q^6 - q^7/( ... ) )))). Also,
A(q) = 1/(1 - q/(1 - (q^2 + q^3)/(1 - q^5/(1 - (q^4 + q^7)/(1 - q^9/(1 - (q^6 + q^11)/(1 - q^13/( ... ) ))))))) and
A(q) = 1/(2 - (1 + q)/(2 - (1 + q^3)/(2 - (1 + q^5)/(2 - (1 + q^7)/( ... ) )))).
O.g.f. as a ratio of q-series: A(q) =  N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2+n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) )^2 and D(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2-n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) )^2.
D(q) has its least (simple) real zero at x = 0.53600 49695 29708 61653 44946 12214 97438 08884 63471 33627....
a(n) ~ c*x^(-n) where c = 0.30516 69461 42293 61432 58334 29163 22891 57284 39056 20388 ... = - N(x)/(x*D'(x)) and the prime indicates differentiation w.r.t. q. (End)

A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 12, 18, 26, 38, 53, 75, 103, 142, 192, 260, 346, 461, 607, 797, 1038, 1348, 1738, 2234, 2856, 3638, 4614, 5832, 7342, 9214, 11525, 14369, 17863, 22142, 27371, 33744, 41498, 50903, 62299, 76066, 92676, 112666, 136696, 165507, 200018

Offset: 0

N. J. A. Sloane

Also n-stacks with strictly receding left wall.
Weakly unimodal compositions such that each up-step is by at most 1 (and first part 1). By dropping the requirement for weak unimodality one obtains A005169. - Joerg Arndt, Dec 09 2012
The values of a(19) and a(20) in Auluck's table on page 686 are wrong (they have been corrected here). - David W. Wilson, Mar 07 2015
Also the number of overpartitions of n having more overlined parts than non-overlined parts.   For example, a(5) = 5 counts the overpartitions [5'], [4',1'], [3',2'], [3',1',1] and [2',2,1']. - Jeremy Lovejoy, Jan 15 2021

			For a(6)=8 we have the following stacks:
..x
.xx .xx. ..xx .x... ..x.. ...x. ....x
xxx xxxx xxxx xxxxx xxxxx xxxxx xxxxx xxxxxx
From _Franklin T. Adams-Watters_, Jan 18 2007: (Start)
For a(7) = 12 we have the following stacks:
..x. ...x
.xx. ..xx .xxx .xx.. ..xx. ...xx
xxxx xxxx xxxx xxxxx xxxxx xxxxx
and
.x.... ..x... ...x.. ....x. .....x
xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxxx
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
J. S. Birman, Letter to N. J. A. Sloane, Apr 09 1994
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Erich Friedman, Illustration of initial terms
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A001522, A001523, A171604, A007293, A015128.

Row sums of triangle A259095.

s := 1+sum(z^(n*(n+1)/2)/((1-z^(n))*product((1-z^i), i=1..n-1)^2), n=1..50): s2 := series(s, z, 300): for j from 1 to 100 do printf(`%d,`,coeff(s2, z, j)) od: # James Sellers, Feb 27 2001
# second Maple program:
b:= proc(n, i) option remember; `if`(i>n, 0, `if`(
      irem(n, i)=0, 1, 0)+add(j*b(n-i*j, i+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2018

m = 45; CoefficientList[ Series[Sum[ z^(n*(n+1)/2)/((1-z^(n))*Product[(1-z^i), {i, 1, n-1}]^2), {n, 1, m}], {z, 0, m}], z] // Prepend[Rest[#], 1] &
(* Jean-François Alcover, May 19 2011, after Maple prog. *)

{a(n) = if( n<0, 0, polcoeff( sum( k=0,(sqrt(8*n + 1) - 1) / 2, x^((k^2 + k) / 2) / prod( i=1, k, (1 - x^i + x * O(x^n))^((iMichael Somos, Apr 27 2003 */

G.f.: sum(n>=1, q^(n*(n+1)/2) / prod(k=1..n-1, 1-q^k)^2 / (1-q^n) ). [Joerg Arndt, Jun 28 2013]
a(n) = sum_{m>0,k>0,2*k^2+k+2*m<=n-1} A008289(m,k)*A000041(n-k*(1+2k)-2*m-1). - [Auluck eq 29]
From Vaclav Kotesovec, Mar 03 2020: (Start)
Pi * sqrt(2/3) <= n^(-1/2)*log(a(n)) <= Pi * sqrt(5/6). [Auluck, 1951]
log(a(n)) ~ 2*Pi*sqrt(n/5). [Wright, 1971]
a(n) ~ exp(2*Pi*sqrt(n/5)) / (sqrt(2) * 5^(3/4) * (1 + sqrt(5)) * n). (End)
a(n) = A143184(n) - A340659(n). - Vaclav Kotesovec, Jun 06 2021

Corrected by R. K. Guy, Apr 08 1988

More terms from James Sellers, Feb 27 2001

A047998 Triangle of numbers a(n,k) = number of "fountains" with n coins, k in the bottom row.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0, 3, 6, 5, 1, 0, 0, 0, 0, 2, 7, 10, 6, 1, 0, 0, 0, 0, 1, 7, 14, 15, 7, 1, 0, 0, 0, 0, 1, 5, 17, 25, 21, 8, 1, 0, 0, 0, 0, 0, 5, 16, 35, 41, 28, 9, 1, 0, 0, 0, 0, 0, 3, 16, 40, 65, 63, 36, 10, 1, 0, 0, 0, 0, 0, 2, 14, 43, 86, 112, 92, 45, 11, 1, 0, 0, 0, 0, 0, 1, 11, 44, 102, 167, 182, 129, 55, 12, 1, 0, 0, 0, 0, 0, 1, 9, 40, 115, 219, 301, 282, 175, 66, 13, 1

Offset: 0

N. J. A. Sloane

The number a(n,k) of (n,k) fountains equals the number of 231-avoiding permutations in the symmetric group S_{k} with exactly n - k inversions (Brändén et al., Proposition 4).

			Triangle begins:
00:  1;
01:  0,1;
02:  0,0,1;
03:  0,0,1,1;
04:  0,0,0,2,1;
05:  0,0,0,1,3,1;
06:  0,0,0,1,3,4,1;
07:  0,0,0,0,3,6,5,1;
08:  0,0,0,0,2,7,10,6,1;
09:  0,0,0,0,1,7,14,15,7,1;
10:  0,0,0,0,1,5,17,25,21,8,1;
11:  0,0,0,0,0,5,16,35,41,28,9,1;
12:  0,0,0,0,0,3,16,40,65,63,36,10,1;
13:  0,0,0,0,0,2,14,43,86,112,92,45,11,1;
14:  0,0,0,0,0,1,11,44,102,167,182,129,55,12,1;
15:  0,0,0,0,0,1,9,40,115,219,301,282,175,66,13,1;
16:  0,0,0,0,0,0,7,37,118,268,434,512,420,231,78,14,1;
17:  0,0,0,0,0,0,5,32,118,303,574,806,831,605,298,91,15,1;
...
From _Joerg Arndt_, Mar 25 2014: (Start)
The compositions (compositions starting with part 1 and up-steps <= 1) corresponding to row n=8 with their base lengths are:
01:    [ 1 2 3 2 ]               4
02:    [ 1 2 2 3 ]               4
03:    [ 1 2 3 1 1 ]             5
04:    [ 1 2 2 2 1 ]             5
05:    [ 1 1 2 3 1 ]             5
06:    [ 1 2 2 1 2 ]             5
07:    [ 1 2 1 2 2 ]             5
08:    [ 1 1 2 2 2 ]             5
09:    [ 1 1 1 2 3 ]             5
10:    [ 1 2 2 1 1 1 ]           6
11:    [ 1 2 1 2 1 1 ]           6
12:    [ 1 1 2 2 1 1 ]           6
13:    [ 1 2 1 1 2 1 ]           6
14:    [ 1 1 2 1 2 1 ]           6
15:    [ 1 1 1 2 2 1 ]           6
16:    [ 1 2 1 1 1 2 ]           6
17:    [ 1 1 2 1 1 2 ]           6
18:    [ 1 1 1 2 1 2 ]           6
19:    [ 1 1 1 1 2 2 ]           6
20:    [ 1 2 1 1 1 1 1 ]         7
21:    [ 1 1 2 1 1 1 1 ]         7
22:    [ 1 1 1 2 1 1 1 ]         7
23:    [ 1 1 1 1 2 1 1 ]         7
24:    [ 1 1 1 1 1 2 1 ]         7
25:    [ 1 1 1 1 1 1 2 ]         7
26:    [ 1 1 1 1 1 1 1 1 ]       8
There are none with base length <= 3, two with base length 4, etc., giving row 8 [0,0,0,0,2,7,10,6,1].
(End)

B. C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991.
R. K. Guy, personal communication to N. J. A. Sloane.
See A005169 for further references.

Alois P. Heinz, Rows n = 0..200, flattened
P. Brändén, A. Claesson, E. Steingrímsson, Catalan continued fractions and increasing subsequences in permutations, Discrete Mathematics, Vol. 258, Issues 1-3, Dec. 2002, 275-287.
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Row sums give A005169 (set x=1 in the g.f.).
Column sums give A000108 (set y=1 in the g.f.). - Joerg Arndt, Mar 25 2014
T(2n+1,n+1) gives A058300(n). - Alois P. Heinz, Jun 24 2015
Cf. A161492.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add(b(n-j, j)*x, j=1..min(i+1, n))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Oct 05 2017

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j]*x, {j, 1, Min[i+1, n]}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jul 11 2018, after Alois P. Heinz *)

N=22; x='x+O('x^N);
G(k)=if (k>N, 1, 1/(1-y*x^k*G(k+1)));
V=Vec( G(1) );
my( N=#V );
rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }
for(n=1, N, print( rvec( V[n]) ) ); \\ print triangle
\\ Joerg Arndt, Mar 25 2014

G.f.: 1/(1 - y*x / (1 - y*x^2 / (1 - y*x^3 / ( ... )))), from the Odlyzko/Wilf reference. - Joerg Arndt, Mar 25 2014

G.f.: ( Sum_{n >= 0} (-y)^n*x^(n*(n+1))/Product_{k = 1..n} (1 - x^k) )/ ( Sum_{n >= 0} (-y)^n*x^(n^2)/Product_{k = 1..n} (1 - x^k) ) = 1 + y*x + y^2*x^2 + (y^2 + y^3)*x^3 + (2*y^3 + y^4)*x^4 + ... (see Berndt, Cor. to Entry 15, ch. 16). - Peter Bala, Jun 20 2019

More terms from Joerg Arndt, Mar 08 2011

A111317 Let f(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is f(q^2,q^3) / f(q,q^3).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, -1, 0, 2, -1, -1, 2, -1, -2, 3, 1, -3, 2, 1, -4, 2, 3, -4, 1, 4, -5, 0, 6, -5, -2, 7, -5, -4, 10, -3, -7, 10, -2, -10, 11, 1, -13, 11, 4, -16, 11, 9, -19, 8, 12, -22, 7, 19, -24, 2, 24, -26, -3, 32, -25, -10, 37, -25, -18, 45, -21, -29, 49, -17, -39, 56, -8, -51, 58, 0, -65, 61, 14, -78, 59, 27, -92

Offset: 0

N. J. A. Sloane, Nov 09 2005

Convolution inverse of A111165.

			From _Peter Bala_, Dec 2012: (Start)
F(1/10) = Sum_{n>=0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(9 + 1/(1 + 1/(99 + 1/(1 + 1/(999 + 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/(101 + 1/(999 + 1/(1001 + ...))))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 25, Equation (39).

Cf. A005169, A143951.

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0, 1, -1][irem(d, 3)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*{0, 1, -1}[[Mod[d, 3]+1]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( prod(k=0, n\3, (1 - x^(3*k+2)) / (1 - x^(3*k+1)), 1 + x * O(x^n)), n))} /* Michael Somos, Dec 23 2007 */

{a(n)=polcoeff(exp(sum(m=1,n+1,1/(1+x^m+x^(2*m)+x*O(x^n))*x^m/m)),n)} \\ Paul D. Hanna, Jan 23 2010

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(-1, -1)
a = EulerTransform(b)
print([a(n) for n in range(88)]) # Peter Luschny, Nov 17 2022

Euler transform of period 3 sequence [ 1, -1, 0, ...]. - Michael Somos, Dec 23 2007
G.f.: Product_{k>=0} (1 - x^(3*k+2)) / (1 - x^(3*k+1)).
G.f.: exp( Sum_{n>=1} 1/(1 + x^n + x^(2n)) * x^n/n ). - Paul D. Hanna, Jan 23 2010
From Peter Bala, Dec 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 2, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-1 + 1/(1 + 1/(n^2-1 + 1/(1 + 1/(n^3-1 + 1/(1 + ...)))))).
For n >= 2, F(-1/n) has the simple continued fraction expansion
1/(1 + 1/(n-1 + 1/(n^2+1 + 1/(n^3-1 + ...)))). Examples are given below. Cf. A005169 and A143951.
(End)

A143951 Number of Dyck paths such that the area between the x-axis and the path is n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 71, 107, 161, 243, 367, 553, 834, 1258, 1898, 2863, 4318, 6514, 9827, 14824, 22361, 33732, 50886, 76762, 115796, 174680, 263509, 397508, 599647, 904579, 1364576, 2058489, 3105269, 4684359, 7066449, 10659877, 16080632, 24257950, 36593598, 55202165, 83273553, 125619799, 189499952

Offset: 0

Emeric Deutsch, Oct 09 2008

nonn

Column sums of A129182.

			a(5)=3 because we have UDUUDD, UUDDUD and UDUDUDUDUD, where U=(1,1) and D=(1,-1).
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/(98 + 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/(10 + 1/(100 + 1/(1000 + ...)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vincenzo Librandi)
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012

Cf. A129182, A291874 (convolution inverse).

Cf. A005169, A111317, A224704.

g:=1/(1-x/(1-x^3/(1-x^5/(1-x^7/(1-x^9/(1-x^11/(1-x^13/(1-x^15)))))))): gser:= series(g,x=0,45): seq(coeff(gser,x,n),n=0..44);
# second Maple program:
b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
      `if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
    end:
a:= n-> add(b(2*n-4*t, 0, n), t=0..n/2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 24 2018

terms = 50; CoefficientList[1/(1+ContinuedFractionK[-x^(2i-1), 1, {i, 1, Sqrt[terms]//Ceiling}]) + O[x]^terms, x] (* Jean-François Alcover, Jul 11 2018 *)

N=66; q = 'q +O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / G(k+2) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013

G.f.: 1/(1 - x/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^9/(1 - ...
Derivation: the g.f. G(x,z) of Dyck paths, where x marks area and z marks semilength, satisfies G(x,z)=1+x*z*G(x,z)*G(x,x^2*z). Set z=1.
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^2-2 + 1/(1 + 1/(n^3-2 + 1/(1 + ...)))))).
For n >= 1, F(-1/n) has the simple continued fraction expansion
1/(1 + 1/(n + 1/(n^2 + 1/(n^3 + ...)))). Examples are given below. Cf. A005169 and A111317.
(End)
G.f.: A(x) = 1/(1 - x/(1-x + x/(1+x^2 + x^4/(1-x^3 - x^2/(1+x^4 - x^7/(1-x^5 + x^3/(1+x^6 + x^10/(1-x^7 - x^4/(1+x^8 - x^13/(1-x^9 + x^5/(1+x^10 + x^16/(1 + ...)))))))))))), a continued fraction. - Paul D. Hanna, Aug 08 2016
a(n) ~ c / r^n, where r = 0.66290148514884371255690407749133031115536799774051... and c = 0.337761150388539773466092171229604432776662930886727976914... . - Vaclav Kotesovec, Feb 17 2017, corrected Nov 04 2021
From Peter Bala, Jul 04 2019: (Start)
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2+n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ) and D(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2-n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ). Cf. A224704.
D(q) has its least  positive (and simple) real zero at x = 0.66290 14851 48843 71255 69040 ....
a(n) ~ c*d^n, where  d = 1/x = 1.5085197761707628638804960 ...  and c = - N(x)/(x*D'(x)) = 0.3377611503885397734660921 ... (the prime indicates differentiation w.r.t. q). (End)

b-file corrected and extended by Alois P. Heinz, Aug 24 2018

A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 4, 4, 6, 7, 8, 11, 13, 16, 20, 24, 31, 37, 46, 58, 70, 88, 108, 133, 167, 204, 252, 315, 386, 479, 594, 731, 909, 1122, 1386, 1720, 2124, 2628, 3254, 4022, 4980, 6160, 7618, 9432, 11665, 14433, 17860, 22093, 27341, 33824, 41847, 51785, 64065, 79267

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Number of rough sandpiles: 1-dimensional sandpiles (see A186085) with n grains without flat steps (no two successive parts of the corresponding composition equal), see example. - Joerg Arndt, Mar 08 2014
The sequence of such sandpiles by base length starts (n>=0) 1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, ... (A097331, essentially A000108 with interlaced zeros).  This is a consequence of the obvious connection to Dyck paths, see example. - Joerg Arndt, Mar 09 2014
a(n>=1) are the Dyck paths with area n between the x-axis and the path which return to the x-axis only once (at their end), whereas A143951 includes paths with intercalated touches of the x-axis. - R. J. Mathar, Aug 22 2018

			From _Joerg Arndt_, Mar 08 2014: (Start)
The a(21) = 7 rough sandpiles are:
:
:   1:      [ 1 2 1 2 1 2 1 2 1 2 3 2 1 ]  (composition)
:
:           o
:  o o o o ooo
: ooooooooooooo  (rendering of sandpile)
:
:
:   2:      [ 1 2 1 2 1 2 1 2 3 2 1 2 1 ]
:
:         o
:  o o o ooo o
: ooooooooooooo
:
:
:   3:      [ 1 2 1 2 1 2 3 2 1 2 1 2 1 ]
:
:       o
:  o o ooo o o
: ooooooooooooo
:
:
:   4:      [ 1 2 1 2 3 2 1 2 1 2 1 2 1 ]
:
:     o
:  o ooo o o o
: ooooooooooooo
:
:
:   5:      [ 1 2 3 2 1 2 1 2 1 2 1 2 1 ]
:
:   o
:  ooo o o o o
: ooooooooooooo
:
:
:   6:      [ 1 2 3 2 3 4 3 2 1 ]
:
:      o
:   o ooo
:  ooooooo
: ooooooooo
:
:
:   7:      [ 1 2 3 4 3 2 3 2 1 ]
:
:    o
:   ooo o
:  ooooooo
: ooooooooo
(End)
From _Joerg Arndt_, Mar 09 2014: (Start)
The A097331(9) = 14 such sandpiles with base length 9 are:
01:  [ 1 2 1 2 1 2 1 2 1 ]
02:  [ 1 2 1 2 1 2 3 2 1 ]
03:  [ 1 2 1 2 3 2 3 2 1 ]
04:  [ 1 2 1 2 3 2 1 2 1 ]
05:  [ 1 2 1 2 3 4 3 2 1 ]
06:  [ 1 2 3 2 1 2 3 2 1 ]
07:  [ 1 2 3 2 1 2 1 2 1 ]
08:  [ 1 2 3 2 3 2 1 2 1 ]
09:  [ 1 2 3 2 3 2 3 2 1 ]
10:  [ 1 2 3 4 3 2 1 2 1 ]
11:  [ 1 2 3 2 3 4 3 2 1 ]
12:  [ 1 2 3 4 3 2 3 2 1 ]
13:  [ 1 2 3 4 3 4 3 2 1 ]
14:  [ 1 2 3 4 5 4 3 2 1 ]
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Joerg Arndt)
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Cf. A049346 (g.f.: 1 - 1/G(0), where G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A173258, A291874.

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 + (-q)^(k+1) / (1 - (-q)^(k+1) / G(k+1) ) );
gf = 1 / G(0);
Vec(gf)

N = 66;  q = 'q + O('q^N);
F(q,y,k) = if(k>N, 1, 1/(1 - y*q^2 * F(q, q^2*y, k+1) ) );
Vec( 1 + q * F(q,q,0) ) \\ Joerg Arndt, Mar 09 2014

a(0) = 1 and a(n) = abs(A049346(n)) for n>=1.
G.f.: 1/ (1-q/(1+q/ (1+q^2/(1-q^2/ (1-q^3/(1+q^3/ (1+q^4/(1-q^4/ (1-q^5/(1+q^5/ (1+-...) )) )) )) )) )).
G.f.: 1 + q * F(q,q) where F(q,y) = 1/(1 - y * q^2 * F(q, q^2*y) ); cf. A005169 and p. 841 of the Odlyzko/Wilf reference; 1/(1 - q * F(q,q)) is the g.f. of A143951. - Joerg Arndt, Mar 09 2014
G.f.: 1 + q/(1 - q^3/(1 - q^5/(1 - q^7/ (...)))) (from formulas above). - Joerg Arndt, Mar 09 2014
G.f.: F(x, x^2) where F(x,y) is the g.f. of A239927. - Joerg Arndt, Mar 29 2014
a(n) ~ c * d^n, where d = 1.23729141259673487395949649334678514763130846902468... and c = 0.0773368373684184197215007198148835507944051447907... - Vaclav Kotesovec, Sep 05 2017
G.f.: A(x) = 2 -1/A143951(x). - R. J. Mathar, Aug 23 2018

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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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A023361 Number of compositions of n into positive triangular numbers.

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A006958 Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused).

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A008795 Molien series for 3-dimensional representation of dihedral group D_6 of order 6.

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A224704 Number of stacks of n triangles, pointing upwards or downwards depending on row parity.

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A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

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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.