A078920 -id:A078920 - cp's OEIS frontend

A136439 Sum of heights of all 1-watermelons with wall of length 2*n.

1, 3, 10, 34, 118, 417, 1495, 5421, 19838, 73149, 271453, 1012872, 3797228, 14294518, 54006728, 204702328, 778115558, 2965409556, 11327549778, 43361526366, 166306579062, 638969153207, 2458973656584, 9477124288144, 36576265716636, 141344492073392, 546860238004919

Offset: 1

Steven Finch, Apr 02 2008

nonn

a(n) is the sum of heights of all Dyck excursions of length 2*n (nonnegative walks beginning and ending at 0 with jumps -1,+1).

N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.

François Marques, Table of n, a(n) for n = 1..1500 (first 650 terms from Alois P. Heinz)
N. Dershowitz and C. Rinderknecht, The Average Height of Catalan Trees by Counting Lattice Paths, Preprint, 2015. Contains more information about the asymptotic behavior than was included in the published version. [Included with permission]
N. Dershowitz and C. Rinderknecht, The Average Height of Catalan Trees by Counting Lattice Paths, Math. Mag., 88 (No. 3, 2015), 187-195.
M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, Elec. J. Combin. 14 (2007) R64.
S. Gilliand, C. Johnson, S. Rush, D. Wood, The sock matching problem, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 691-697; DOI: 10.2140/involve.2014.7.691.

Cf. A000108, A008549, A078920, A261003, A333498.

H[0]:=1: for k to 30 do H[k]:=simplify(1/(1-z*H[k-1])) end do: g:=sum(j*(H[j]-H[j-1]),j=1..30): gser:=series(g,z=0,27): seq(coeff(gser,z,n),n=1..24); # Emeric Deutsch, Apr 13 2008
# second Maple program:
b:= proc(x, y, h) option remember; `if`(x=0, h, add(`if`(x+j>y,
      b(x-1, y-j, max(h, y-j)), 0), j={$-1..min(1, y)} minus {0}))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=1..33);  # Alois P. Heinz, Mar 24 2020

c[n_] := (2*n)!/(n!*(n+1)!)
s[n_,a_] := Sum[If[k < 1, 0, DivisorSigma[0,k]*Binomial[2*n,n+a-k]/Binomial[2*n,n]], {k,a-n,a+n}]
h[n_] := (n+1)*(s[n,1]-2*s[n,0]+s[n,-1]) - 1
a[n_] := h[n]*c[n]

\\ Translation of Mathematica code
s(n,a)=sum(k=1,a+n, numdiv(k)*binomial(2*n,n+a-k))/binomial(2*n,n)
a(n)=((n+1)*(s(n,1)-2*s(n,0)+s(n,-1))-1)*binomial(2*n,n)/(n+1) \\ Charles R Greathouse IV, Mar 28 2016

G.f.: Sum_{k >= 1} k*(H[k]-H[k-1]), where H[0]=1 and H[k]=1/(1-zH[k-1]) for k=1,2,... (the first Maple program makes use of this g.f.). - Emeric Deutsch, Apr 13 2008

More terms from Alois P. Heinz, Mar 24 2020

A181119 Number of transpose-complementary plane partitions of n.

Original entry on oeis.org

1, 2, 84, 81796, 1844536720, 962310111888300, 11608208114358751650000, 3236574482779383546336417240000, 20853456581643133066208521560263633137920, 3104385823530881109001458753652585998600603921849920, 10676554307318599842868990948461304923921623250562199975300214736

Offset: 0

Arvind Ayyer, Jan 21 2011

The complement of a plane partition inside an m X m X m cube consists of the boxes which are within the cube, but not in the plane partition, rotated in an appropriate way.

a(n) is the number of plane partitions inside an 2n X 2n X 2n cube whose (matrix) transpose when written as an 2n X 2n array is the same as its complement.

			When n=2, there are two transpose-complementary plane partitions,
[1 1] and [2 1], both of whose transpose and complement is equal to themselves.
[1 1]     [1 0]

R. P. Stanley, Symmetries of Plane Partitions, J. Comb. Theory Ser. A 43 (1986), 103-113.
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
Wikipedia, Plane partition

Cf. A008793, A051255, A078920, A123352.

Table[Binomial[3n-1,n]Product[(2n+i+j+1)/(i+j+1),{i,1,2n-2}, {j,i,2n-2}], {n,0,10}] (* Harvey P. Dale, Jan 27 2012 *)

a(n) = binomial(3*n-1,n)*prod(i=1,2*n-2,prod(j=i,2*n-2,(2*n+i+j+1)/(i+j+1))); \\ Michel Marcus, Jun 18 2015

a(n) = binomial(3n-1,n)*Product(i=1..2n-2,Product(j=i..2n-2,(2n+i+j+1)/(i+j+1))).

a(n) ~ exp(1/24) * 3^(9*n^2 - 3*n/2 - 1/24) / (sqrt(A) * n^(1/24) * 2^(12*n^2 - n - 1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 28 2015

A355503 Total number of m-tuples (p_1, p_2, ..., p_m) of Dyck paths of semilength n-m, such that each p_i is never below p_{i-1} for m=0..n.

Original entry on oeis.org

1, 2, 3, 5, 11, 35, 164, 1120, 10969, 152849, 3029650, 85227078, 3400752392, 192644205130, 15470939367651, 1761760468965521, 284641456742538865, 65175288287611738435, 21159611204475209730138, 9743708333490185603430830, 6357930817596444858142966826

Offset: 0

Alois P. Heinz, Jul 04 2022

nonn

			a(3) = 5: ( ), (/\/\), (//\\), (/\, /\, /\), (<>, <>, <>, <>).

Alois P. Heinz, Table of n, a(n) for n = 0..115
Wikipedia, Counting lattice paths

Cf. A000108, A074962, A078920, A123352, A355400.

Antidiagonal sums of A368025.

a:= n-> add(mul(mul((i+j+2*(n-m))/(i+j), j=i..m-1), i=1..m-1), m=0..n):
seq(a(n), n=0..23);

Table[Sum[Product[Product[(i+j+2*(n-m))/(i+j), {j,i,m-1}], {i,1,m-1}], {m,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)
Table[Sum[BarnesG[1 + m] * Sqrt[BarnesG[1 + 2*n] * BarnesG[2 - 2*m + 2*n] * Gamma[1 + 2*m] * Gamma[1 + n] / (BarnesG[1 + 2*m] * Gamma[1 + m] * Gamma[1 + 2*n] * Gamma[1 - m + n])] / BarnesG[1 - m + 2*n], {m, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2023 *)

a(n) = Sum_{m=0..n} Product_{i=1..m-1, j=i..m-1} (i+j+2*(n-m))/(i+j).
a(n) = 1 + Sum_{k=0..n-1} A078920(n-1,k).
a(n) = 1 + Sum_{k=0..n-1} A123352(n-1,k).
a(n) = Sum_{k=0..n} A368025(n-k, k).
From Vaclav Kotesovec, Aug 27 2023: (Start)
a(n) ~ c * exp(1/24) * 3^(n^2 - n/2) / (sqrt(A) * n^(1/24) * 2^((4*n^2-n-1)/3)), where A = A074962 is the Glaisher-Kinkelin constant and
c = Sum_{k,-oo,oo} 2^((k + mod(n,3)/3)/2 - 3*(k + mod(n,3)/3)^2/2).
Numerically, c = 1.78933741155287907159762028... if mod(n,3)=0 or mod(n,3)=1 and c = 1.78893263307672974352375161... if mod(n,3)=2. (End)

A136440 Sum of heights of all 2-watermelons with wall of length 2*n.

Original entry on oeis.org

3, 11, 60, 406, 3171, 27411, 255617, 2528613, 26224097, 282706396, 3147801820, 36022733951, 422047425238, 5046771514478, 61438059222438, 759851375725606, 9530872096367508, 121063493728881999, 1555352365759798758

Offset: 1

Steven Finch, Apr 02 2008

nonn

Consider the set of all pairs of nonintersecting Dyck excursions of length 2*n (nonnegative walks with jumps -1,+1). The lower path begins and ends at 0; the upper path begins and ends at 2. a(n) is the sum of heights of all such upper-Dyck excursions.

M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, Elec. J. Combin. 14 (2007) R64.

Cf. A005700, A078920.

c[n_] := 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!)
s[n_,a_] := Sum[If[k < 1, 0, DivisorSigma[0,k]*Binomial[2*n,n+a-k]/Binomial[2*n,n]], {k,a-n,a+n}]
t[n_,a_,b_] := Sum[If[(j < 1) || (k < 1), 0, DivisorSigma[0,GCD[j,k]]*Binomial[2*n,n+a-j]*Binomial[2*n,n+b-k]/Binomial[2*n,n]^2], {j,a-n,a+n}, {k,b-n,b+n}]
f[n_] := (n^2+5*n+6)*(s[n,-3]+s[n,3])-(6*n^2+18*n)*(s[n,-2]+s[n,2])+(15*n^2+27*n+6)*(s[n,-1]+s[n,1])-(20*n^2+28*n+24)*s[n,0]
g[n_] := t[n,-2,-2]-t[n,-1,-3]-2*t[n,-1,-2]+t[n,-1,-1]+2*t[n,-1,0]-t[n,-1,3]+2*t[n,0,-3]-4*t[n,0,0]+2*t[n,0,3]-t[n,1,-3]-2*t[n,1,-2]+2*t[n,1,-1]+2*t[n,1,0]+t[n,1,1]-t[n,1,3]+2*t[n,2,-2]-2*t[n,2,-1]-2*t[n,2,1]+t[n,2,2]
h[n_] := ((n+1)*(n+2)/(12*(2*n+1)))*( (n+1)*(n+2)*(n+3)*g[n]+f[n] ) - 1
a[n_] := h[n]*c[n]

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A136439 Sum of heights of all 1-watermelons with wall of length 2*n.

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A181119 Number of transpose-complementary plane partitions of n.

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A355503 Total number of m-tuples (p_1, p_2, ..., p_m) of Dyck paths of semilength n-m, such that each p_i is never below p_{i-1} for m=0..n.

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A136440 Sum of heights of all 2-watermelons with wall of length 2*n.

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