A005169 -id:A005169 - cp's OEIS frontend

A003504 a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).

1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856

Offset: 0

N. J. A. Sloane, R. K. Guy

The sequence appears with a different offset in some other sources. - Michael Somos, Apr 02 2006
Also known as Göbel's (or Goebel's) Sequence. Asymptotically, a(n) ~ n*C^(2^n) where C=1.0478... (A115632). A more precise asymptotic formula is given in A116603. - M. F. Hasler, Dec 12 2007
Let s(n) = (n-1)*a(n). By considering the p-adic representation of s(n) for primes p=2,3,...,43, one finds that a(44) is the first nonintegral value in this sequence. Furthermore, for n>44, the valuation of s(n) w.r.t. 43 is -2^(n-44), implying that both s(n) and a(n) are nonintegral. - M. F. Hasler and Max Alekseyev, Mar 03 2009
a(44) is approximately 5.4093*10^178485291567. - Hans Havermann, Nov 14 2017.
The fractional part is simply 24/43 (see page 709 of Guy (1988)).
The more precise asymptotic formula is a(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...). - Michael Somos, Mar 17 2012

			a(3) = (1 * 2 + 2^2) / 2 = 3 given a(2) = 2.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Sect. E15.
Clifford Pickover, A Passion for Mathematics, 2005.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..16
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 1.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
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]
H. Ibstedt, Some sequences of large integers, Fibonacci Quart. 28 (1990), 200-203.
Yuh Kobayashi and Shin-ichiro Seki, On the length over which k-Göbel sequences remain integers, arXiv:2502.17448 [math.CO], 2025.
H. W. Lenstra, Jr., R. K. Guy, and N. J. A. Sloane, Correspondence, 1975-1978
N. Lygeros and M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)
Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, How long can k-Göbel sequences remain integers?, arXiv:2307.09741 [math.NT], 2023.
D. Rusin, Law of small numbers [Broken link]
D. Rusin, Law of small numbers [Cached copy]
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Eric Weisstein's World of Mathematics, Göbel's Sequence
D. Zagier, Problems posed at the St Andrews Colloquium, 1996
D. Zagier, Solution: Day 5, problem 3

Cf. A005166, A005167, A097398, A108394, A115632, A116603 (asymptotic formula).

a:=2: L:=1,1,a: n:=15: for k to n-2 do a:=a*(a+k)/(k+1): L:=L,a od:L; # Robert FERREOL, Nov 07 2015

a[n_] := a[n] = Sum[a[k]^2, {k, 0, n-1}]/(n-1); a[0] = a[1] = 1; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 06 2013 *)
With[{n = 14}, Nest[Append[#, (#.#)/(Length[#] - 1)] &, {1, 1}, n - 2]] (* Jan Mangaldan, Mar 21 2013 *)

A003504(n,s=2)=if(n-->0,for(k=1,n-1,s+=(s/k)^2);s/n,1) \\ M. F. Hasler, Dec 12 2007

a=2; L=[1,1,a]; n=15
for k in range(1,n-1):
    a=a*(a+k)//(k+1)
    L.append(a)
print(L) # Robert FERREOL, Nov 07 2015

a(n+1) = ((n-1) * a(n) + a(n)^2) / n if n > 1. - Michael Somos, Apr 02 2006

0 = a(n)*(+a(n)*(a(n+1) - a(n+2)) - a(n+1) - a(n+1)^2) +a(n+1)*(a(n+1)^2 - a(n+2)) if n>1. - Michael Somos, Jul 25 2016

a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral - H. W. Lenstra, Jr.
Corrected and extended by M. F. Hasler, Dec 12 2007
Further corrections from Max Alekseyev, Mar 04 2009

A053260 Coefficients of the '5th-order' mock theta function psi_0(q).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 8, 9, 10, 9, 11, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 29, 30, 32, 32, 34, 36, 36, 39, 40, 41, 44, 45, 46

Offset: 0

Dean Hickerson, Dec 19 1999

Number of partitions of n such that each part occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive integers occur.

Strongly unimodal compositions with first part 1 and each up-step is by at most 1 (left-smoothness); with this interpretation one should set a(0)=1; see example. Replacing "strongly" by "weakly" in the condition gives A001524. Dropping the requirement of unimodality gives A005169. [Joerg Arndt, Dec 09 2012]

			From _Joerg Arndt_, Dec 09 2012: (Start)
The a(42)=8 strongly unimodal left-smooth compositions are
[ #]       composition
[ 1]    [ 1 2 3 4 5 6 7 5 4 3 2 ]
[ 2]    [ 1 2 3 4 5 6 7 6 4 3 1 ]
[ 3]    [ 1 2 3 4 5 6 7 6 5 2 1 ]
[ 4]    [ 1 2 3 4 5 6 7 6 5 3 ]
[ 5]    [ 1 2 3 4 5 6 7 8 3 2 1 ]
[ 6]    [ 1 2 3 4 5 6 7 8 4 2 ]
[ 7]    [ 1 2 3 4 5 6 7 8 5 1 ]
[ 8]    [ 1 2 3 4 5 6 7 8 6 ]
(End)

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, i-1))))
    end:
a:= proc(n) local h, k, m, r;
      m, r:= floor((sqrt(n*8+1)-1)/2), 0;
      for k from m by -1 do h:= k*(k+1);
        if h<=n then break fi;
        r:= r+b(n-h/2, k-1)
      od: r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 02 2013

Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-1] ] ]]; a[n_] := Module[{h, k, m, r}, {m, r} = {Floor[(Sqrt[n*8+1]-1)/2], 0}; For[k = m, True, k--, h = k*(k+1); If[h <= n, Break[]]; r = r + b[n-h/2, k-1]]; r]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, x^(n*(n+1)/2) * prod(k=1,n-1,1+x^k) ) + 'c0;
v = Vec(gf); v[1]-='c0; v
/* Joerg Arndt, Apr 21 2013 */

G.f.: psi_0(q) = Sum_{n>=0} q^((n+1)*(n+2)/2) * (1+q)*(1+q^2)*...*(1+q^n).

a(n) ~ exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 0, 1, 1, 1, 2, 3, 4, 5, 1, -1, 1, 1, 1, 2, 3, 5, 7, 6, 1, 0, 1, 1, 1, 2, 3, 5, 8, 11, 10, 1, 0, 1, 1, 1, 2, 3, 5, 9, 13, 17, 14, 1, 0, 1, 1, 1, 2, 3, 5, 9, 14, 22, 28, 21, 1, 0

Offset: 0

Alois P. Heinz, Aug 29 2013

			Square array A(n,m) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  2,  2,  2,  2,  2,  2,  2, ...
   0,  1,  2,  3,  3,  3,  3,  3,  3, ...
  -1,  1,  3,  4,  5,  5,  5,  5,  5, ...
   0,  1,  5,  7,  8,  9,  9,  9,  9, ...
   0,  1,  6, 11, 13, 14, 15, 15, 15, ...
  -1,  1, 10, 17, 22, 24, 25, 26, 26, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013

Columns m=0-10 give: A143064, A000012, A227360, A173173(n+1), A227374, A227375, A228646, A228644, A185648, A228645, A185649.

Diagonal gives: A005169.

nMax = 12; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A = Table[col[m][[1 ;; nMax + 1]], {m, 0, nMax}] // Transpose; a[n_ /; 0 <= n <= nMax, m_ /; 0 <= m <= nMax] := With[{n1 = n + 1, m1 = m + 1}, A[[n1, m1]]]; Table[a[n - m, m], {n, 0, nMax}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2016 *)

A186085 Number of 1-dimensional sandpiles with n grains.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 100, 166, 277, 461, 769, 1282, 2137, 3565, 5945, 9916, 16540, 27589, 46022, 76769, 128062, 213628, 356366, 594483, 991706, 1654352, 2759777, 4603843, 7680116, 12811951, 21372882, 35654237, 59478406, 99221923, 165522118, 276124217, 460630839

Offset: 0

Joerg Arndt, Feb 12 2011

Number of compositions of n where the first and the last parts are 1 and the absolute difference between consecutive parts is <=1 (smooth compositions).
Such a composition [c1,c2,c3,...] corresponds to a sandpile with c1(=1) grains in the first positions, c2 in the second, and so on.  Assuming the critical slope is 1 (for the pile to be stable) we obtain the conditions on the compositions.
With the additional requirement of unimodality one gets A001522. [Joerg Arndt, Dec 09 2012]
Dropping the requirement that the first and last parts are 1 gives A034297. Restriction to weakly increasing (or decreasing) sums gives A034296. [Joerg Arndt, Jun 02 2013]
Also the number of compositions of n with first part 1, up-steps of at most 1, and no two consecutive up-steps.  The sandpiles are recovered by shifting the rows above the bottom row to the left by one position relative to the next lower row. [Joerg Arndt, Mar 30 2014]
Also fountains of coins (cf. A005169) with no consecutive up-steps. Shift the top rows in the previous comment by half a position. [Joerg Arndt, Mar 30 2014]

			The a(7)=8 smooth compositions of 7 are:
:   1:      [ 1 1 1 1 1 1 1 ]  (composition)
:
: ooooooo  (rendering of sandpile)
:
:   2:      [ 1 1 1 1 2 1 ]
:
:     o
: oooooo
:
:   3:      [ 1 1 1 2 1 1 ]
:
:    o
: oooooo
:
:   4:      [ 1 1 2 1 1 1 ]
:
:   o
: oooooo
:
:   5:      [ 1 1 2 2 1 ]
:
:   oo
: ooooo
:
:   6:      [ 1 2 1 1 1 1 ]
:
:  o
: oooooo
:
:   7:      [ 1 2 1 2 1 ]
:
:  o o
: ooooo
:
:   8:      [ 1 2 2 1 1 ]
:
:  oo
: ooooo

Seiichi Manyama, Table of n, a(n) for n = 0..4502 (terms 0..1000 from Alois P. Heinz)

Cf. A186084 (sandpiles by base length).
Cf. A005169 (compositions of n with c(1)=1 and c(i+1)<=c(i)+1).
Cf. A186505 (antidiagonal sums of triangle A186084).
Cf. A001522, A001523, A001524.
Cf. A129181.

b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, add(b(n-i, i+j), j=-1..1)))
    end:
a:= n-> `if`(n=0, 1, b(n-1, 1)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 11 2013

b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Sum[b[n-i, i+j], {j, -1, 1}]]]; a[n_] := If[n == 0, 1, b[n-1, 1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)

{a(n)=local(Txy=1+x*y); for(i=1, n, Txy=1/(1-x*y-x^3*y^2*subst(Txy, y, x*y+x*O(x^n)))); polcoeff(subst(1+x*Txy, y, 1), n, x)} /* Paul D. Hanna */

/* continued fraction for terms up to 460630839: */
Vec(1/ (1-x/ (1-x^3/ (1-x^2/ (1-x^3/ (1-x^7/ (1-x^4/ (1-x^5/ (1-x^11/ (1-x^6/(1-x*O(x^0) ))))))))))) /* Paul D. Hanna */

N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) );
gf = 1 + 1/Q(0);
Vec(gf) /* Joerg Arndt, May 07 2013 */

G.f.: 1 + x/(1-x - x^3*B(x)) where B(x) equals the g.f. of the antidiagonal sums of triangle A186084 [Paul D. Hanna].
G.f.: 1 + x/(1-x - x^3/(1-x^2 - x^5/(1-x^3 - x^7/(1-x^4 - x^9/(1 -...))))) (continued fraction).  [Paul D. Hanna].
G.f.: 1/(1 - x/(1-x^3/(1-x^2/(1 - x^3/(1-x^7/(1-x^4/(1 - x^5/(1-x^11/(1-x^6/(1 -...)))))))))) (continued fraction).  [Paul D. Hanna].
The g.f. T(x,y) of triangle A186084 satisfies: T(x,y) = 1/(1 - x*y - x^3*y^2*T(x,x*y)); therefore, the g.f. of this sequence is A(x) = 1 + x*T(x,1). [Paul D. Hanna]
a(n) ~ c/r^n where r = 0.5994477646147968266874606710272382... and c = 0.213259838728143595595398989847345... [Paul D. Hanna]
G.f.: 1 + 1/Q(0), where Q(k)= 1/x^(k+1) - 1 - 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013
G.f.: G(1), where G(k) = 1 + x^k/( 1 - x^k * G(k+1) ) (continued fraction). [Joerg Arndt, Jun 29 2013]
a(n) = Sum_{j=1..n} A129181(n-j,j-1) for n>=1. - Alois P. Heinz, Jun 25 2023

A192728 G.f. satisfies: A(x) = 1/(1 - xA(x)/(1 - x^2A(x)/(1 - x^3A(x)/(1 - x^4A(x)/(1 - ...))))), a recursive continued fraction.

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 226, 822, 3061, 11615, 44746, 174552, 688122, 2737153, 10972066, 44279234, 179754362, 733554695, 3007551211, 12382623614, 51174497023, 212218265661, 882810782322, 3682922292680, 15404800893438, 64590512696020, 271425803359505

Offset: 0

Paul D. Hanna, Jul 08 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 226*x^6 +...
which satisfies A(x) = P(x)/Q(x) where
P(x) = 1 - x^2*A(x)/(1-x) + x^6*A(x)^2/((1-x)*(1-x^2)) - x^12*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^20*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
Q(x) = 1 - x*A(x)/(1-x) + x^4*A(x)^2/((1-x)*(1-x^2)) - x^9*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^16*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
Explicitly, the above series begin:
P(x) = 1 - x^2 - 2*x^3 - 4*x^4 - 10*x^5 - 28*x^6 - 90*x^7 - 310*x^8 - 1114*x^9 - 4115*x^10 - 15522*x^11 - 59517*x^12 - 231284*x^13 +...
Q(x) = 1 - x - 2*x^2 - 4*x^3 - 9*x^4 - 26*x^5 - 84*x^6 - 292*x^7 - 1054*x^8 - 3908*x^9 - 14774*x^10 - 56742*x^11 - 220778*x^12 - 868452*x^13 +...
Also, the g.f. A = A(x) satisfies:
A = 1 + x*A + x^2*A^2 + x^3*(A^3 + A^2) + x^4*(A^4 + 2*A^3) + x^5*(A^5 + 3*A^4 + A^3) + x^6*(A^6 + 4*A^5 + 3*A^4 + A^3) + x^7*(A^7 + 5*A^6 + 6*A^5 + 3*A^4) +...
which is a series generated by the continued fraction expression.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Cf. A005169, A192729, A192730.

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

/* By Ramanujan's continued fraction identity: */
{a(n)=local(A=1+x,P,Q);for(i=1,n,
P=sum(m=0,sqrtint(n),x^(m*(m+1))/prod(k=1,m,1-x^k)*(-A+x*O(x^n))^m);
Q=sum(m=0,sqrtint(n),x^(m^2)/prod(k=1,m,1-x^k)*(-A+x*O(x^n))^m);A=P/Q);polcoeff(A,n)}

G.f. satisfies: A(x) = P(x)/Q(x) where
  P(x) = Sum_{n>=0} x^(n*(n+1)) * (-A(x))^n / Product_{k=1..n} (1-x^k),
  Q(x) = Sum_{n>=0} x^(n^2) * (-A(x))^n / Product_{k=1..n} (1-x^k),
due to Ramanujan's continued fraction identity.
a(n) ~ c * d^n / n^(3/2), where d = 4.44776682810490219629673157389741... and c = 0.533241700941579126635423052024... - Vaclav Kotesovec, Apr 30 2017

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/( ... )))))).

A230267 Coins left after packing 5 curves coins patterns into fountain of coins base n.

Original entry on oeis.org

1, 3, 2, 6, 7, 9, 12, 16, 17, 23, 26, 30, 35, 41, 44, 52, 57, 63, 70, 78, 83, 93, 100, 108, 117, 127, 134, 146, 155, 165, 176, 188, 197, 211, 222, 234, 247, 261, 272, 288, 301, 315, 330, 346, 359, 377, 392, 408, 425, 443

Offset: 1

Kival Ngaokrajang, Oct 15 2013

nonn

Refer to arrangement same as 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". The 5 curves coins patterns consist of a part of circumference and forms continuous area. There is total 13 distinct patterns. I would like to call "5C4S" type as it cover 4 coins and symmetry. When packing 5C4S into fountain of coins base n, the total number of 5C4S is A001399, the coins left is a(n) and void is A230276. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (U)

Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).

G.f.: x*(x^3 - 2*x^2 + 2*x + 1)/((1-x)*(1-x^2)*(1-x^3)) (conjectured). - Ralf Stephan, Oct 17 2013

A230276 Voids left after packing 5-curves coins patterns into fountain of coins with base n.

Original entry on oeis.org

0, 1, 1, 6, 10, 16, 24, 34, 43, 57, 70, 85, 102, 121, 139, 162, 184, 208, 234, 262, 289, 321, 352, 385, 420, 457, 493, 534, 574, 616, 660, 706, 751, 801, 850, 901, 954, 1009, 1063, 1122, 1180, 1240, 1302, 1366, 1429

Offset: 1

Kival Ngaokrajang, Oct 15 2013

Refer to arrangement same as 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". The 5 curves coins patterns consist of a part of each coin circumference and forms a continuous area. There are total 13 distinct patterns. For selected pattern, I would like to call "5C4S" type as it cover 4 coins and symmetry. When packing 5C4S into fountain of coins base n, the total number of 5C4S is A001399, the coins left is A230267 and void left is a(n). See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).

A099837 := proc(n)
    op(modp(n,3)+1,[2,-1,-1]) ;
end proc:
A230276 := proc(n)
    -A099837(n)/3 + (-48*n+31+18*n^2+9*(-1)^n)/24 ;
end proc:
seq(A230276(n),n=1..40) ; # R. J. Mathar, Feb 28 2018

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 1, 1, 6, 10, 16}, 45] (* Jean-François Alcover, May 05 2023 *)

G.f.: x^2*(x^4 + 3*x^3 + 4*x^2 + 1)/((1-x)*(1-x^2)*(1-x^3)). - Ralf Stephan, Oct 17 2013

a(n) = (9*(-1)^n+18*n^2-48*n)/24 - A099837(n)/3. - R. J. Mathar, Feb 28 2018

A129182 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 7, 0, 7, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 14, 0, 17, 0, 16, 0, 16, 0, 14, 0, 11, 0, 9, 0, 7, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 0

Emeric Deutsch, Apr 08 2007

Row n has n^2 + 1 terms.
Row sums are the Catalan numbers (A000108).
Sum(k*T(n,k), k=0..n^2) = A008549(n).
Sums along falling diagonals give A005169. - Joerg Arndt, Mar 29 2014
T(2n,4n) = A240008(n). - Alois P. Heinz, Mar 30 2014

			T(4,10) = 3 because we have UDUUUDDD, UUUDDDUD and UUDUDUDD.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,2,0,1,0,1;
0,0,0,0,1,0,3,0,3,0,3,0,2,0,1,0,1;
0,0,0,0,0,1,0,4,0,6,0,7,0,7,0,5,0,5,0,3,0,2,0,1,0,1;
Transposed triangle (A239927) 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;
... - _Joerg Arndt_, Mar 25 2014

Alois P. Heinz, Rows n = 0..32, flattened

Cf. A000108, A008549, A139262, A240008, A143951 (column sums).

G:=1/(1-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G,z=0,11)): for n from 0 to 7 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
       expand(b(x-1, y-1)*z^(y-1/2)+ b(x-1, y+1)*z^(y+1/2))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 29 2014

b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2) + b[x-1, y+1]*z^(y+1/2)]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

G.f.: G(t,z) given by G(t,z) = 1+t*z*G(t,t^2*z)*G(t,z).

Sum_{k=0..n^2} (n^2-k)/2 * T(n,k) = A139262(n). - Alois P. Heinz, Mar 31 2018

A161492 Triangle T(n,m) read by rows: the number of skew Ferrers diagrams with area n and m columns.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 17, 13, 5, 1, 1, 12, 32, 34, 19, 6, 1, 1, 16, 55, 78, 58, 26, 7, 1, 1, 20, 89, 160, 154, 90, 34, 8, 1, 1, 25, 136, 305, 365, 269, 131, 43, 9, 1, 1, 30, 200, 544, 794, 716, 433, 182, 53, 10, 1, 1, 36, 284, 923, 1609, 1741, 1271, 657, 244, 64, 11, 1

Offset: 1

R. J. Mathar, Jun 11 2009

Row sums give A006958, sums along falling diagonals give A227309. [Joerg Arndt, Mar 23 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. T(n,m) equals the number of coin fountains with exactly n coins in the even-numbered rows and n - m coins in the odd-numbered rows of the fountain. See the illustration in the Links section. - Peter Bala, Jul 21 2019

			T(4,2)=4 counts the following 4 diagrams with area equal to 4 and 2 columns:
   .X..XX...X..XX
   XX..XX...X..X.
   X.......XX..X.
From _Joerg Arndt_, Mar 23 2014: (Start)
Triangle begins:
01:  1
02:  1   1
03:  1   2    1
04:  1   4    3    1
05:  1   6    8    4     1
06:  1   9   17   13     5     1
07:  1  12   32   34    19     6     1
08:  1  16   55   78    58    26     7    1
09:  1  20   89  160   154    90    34    8   1
10:  1  25  136  305   365   269   131   43   9   1
11:  1  30  200  544   794   716   433  182  53  10  1
12:  1  36  284  923  1609  1741  1271  657 ...
(End)

Peter Bala, Illustration for the terms of row 4
Peter Bala, Fountains of coins and skew Ferrers diagrams
M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers diagrams, Disc. Math. (1993) Vol.112, No. 1-3, pp. 65-79.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See p. 19.
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. 5.

Row sums A006958. Cf. A005169, A227309.

qpoch := proc(a,q,n)
    mul( 1-a*q^k,k=0..n-1) ;
end proc:
A161492 := proc(n,m)
    local N,N2,ns ;
    N := 0 ;
    for ns from 0 to n+2 do
        N := N+ (-1)^ns *q^binomial(ns+1,2) / qpoch(q,q,ns) / qpoch(q,q,ns+1) *q^(ns+1) *t^(ns+1) ;
        N := taylor(N,q=0,n+1) ;
    end do:
    N2 := 0 ;
    for ns from 0 to n+2 do
        N2 := N2+ (-1)^ns*q^binomial(ns,2)/(qpoch(q,q,ns))^2*q^ns*t^ns ;
        N2 := taylor(N2,q=0,n+1) ;
    end do:
    coeftayl(N/N2,q=0,n) ;
    coeftayl(%,t=0,m) ;
end proc:
for a from 1 to 20 do
    for c from 1 to a do
        printf("%d ", A161492(a,c)) ;
    od:
od:

nmax = 13;
qn[n_] := Product[1 - q^k, {k, 1, n}];
nm = Sum[(-1)^n q^(n(n+1)/2)/(qn[n] qn[n+1])(t q)^(n+1) + O[q]^nmax, {n, 0, nmax}];
dn = Sum[(-1)^n q^(n(n-1)/2)/(qn[n]^2)(t q)^n + O[q]^nmax, {n, 0, nmax}];
Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[nm/dn, q]] // Flatten (* Jean-François Alcover, Dec 19 2019, after Joerg Arndt *)

/* formula from the Delest/Fedou reference: */
N=20;  q='q+O('q^N);  t='t;
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 );
v=Vec(nm/dn);
for(n=1,N-1,print(Vec(polrecip(Pol(v[n])))));  \\ print triangle
\\ Joerg Arndt, Mar 23 2014

From Peter Bala, Jul 21 2019: (Start)
The following formulas all include an initial term T(0,0) = 1.
O.g.f. as a ratio of q-series: A(q,t) = N(q,t)/D(q,t) = 1 + q*t + q^2*(t + t^2) + q^3*(t + 2*t^2 + t^3) + ..., where N(q,t) = Sum_{n >= 0} (-1)^n*q^((n^2 + 3*n)/2)*t^n/Product_{k = 1..n} (1 - q^k)^2 and D(q,t) = Sum_{n >= 0} (-1)^n*q^((n^2 + n)/2)*t^n/Product_{k = 1..n} (1 - q^k)^2.
Continued fraction representations:
A(q,t) = 1/(1 - q*t/(1 - q/(1 - q^2*t/(1 - q^2/(1 - q^3*t/(1 - q^3/(1 - (...) ))))))).
A(q,t) = 1/(1 - q*t/(1 + q*(t - 1) - q*t/(1 + q*(t - q) - q*t/( 1 + q*(t - q^2) - q*t/( (...) ))))).
A(q,t) = 1/(1 - q*t - q^2*t/(1 - q*(1 + q*t) - q^4*t/(1 - q^2*(1 + q*t) - q^6*t/(1 - q^3*(1 + q*t) - q^8*t/( (...) ))))).
A(q,t) =  1/(1 - q*t - q^2*t/(1 - q^2*t - q/(1 - q^3*t - q^5*t/(1 - q^4*t - q^2/(1 - q^5*t - q^8*t/ (1 - q^6*t - q^3/(1 - q^7*t - q^11*t/(1 - q^8*t - (...) )))))))). (End)

cp's OEIS Frontend

A003504 a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).

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A053260 Coefficients of the '5th-order' mock theta function psi_0(q).

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A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))).

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A186085 Number of 1-dimensional sandpiles with n grains.

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A192728 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^2*A(x)/(1 - x^3*A(x)/(1 - x^4*A(x)/(1 - ...))))), a recursive continued fraction.

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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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A230267 Coins left after packing 5 curves coins patterns into fountain of coins base n.

A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))).

A192728 G.f. satisfies: A(x) = 1/(1 - xA(x)/(1 - x^2A(x)/(1 - x^3A(x)/(1 - x^4A(x)/(1 - ...))))), a recursive continued fraction.