OFFSET
1,5
COMMENTS
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
LINKS
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, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 5.
FORMULA
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)
EXAMPLE
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)
MAPLE
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:
MATHEMATICA
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 *)
PROG
(PARI) /* 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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Jun 11 2009
STATUS
approved