A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 1, 1, 1, 75, 409, 63, 1, 1, 1, 541, 23917, 16081, 321, 1, 1, 1, 4683, 2244361, 10681263, 699121, 1683, 1, 1, 1, 47293, 308682013, 14638956721, 5552351121, 32193253, 8989, 1, 1, 1, 545835, 58514835289, 35941784497263, 117029959485121, 3147728203035, 1538743249, 48639, 1, 1

Offset: 0

Author

Alois P. Heinz, Oct 02 2015

Keywords

• nonn
• tabl

Comments

Also, A(n,k) is the number of alignments for k sequences of length n each (Slowinski 1998).

Row r > 0 is asymptotic to sqrt(r*Pi) * (r^(r-1)/(r-1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)), or equivalently to sqrt(r) * (r^(r-1)/(r-1)!)^n * (n!)^r / (2^r * (Pi*n)^((r-1)/2) * (log(2))^(r*n+1)). - Vaclav Kotesovec, Mar 23 2016

From Vaclav Kotesovec, Mar 23 2016: (Start)

Column k > 0 is asymptotic to sqrt(c(k)) * d(k)^n / (Pi*n)^((k-1)/2), where c(k) and d(k) are roots of polynomial equations of degree k, independent on n.

---------------------------------------------------

k d(k)

---------------------------------------------------

2 5.8284271247461900976033774484193...

3 56.9476283720414911685286267804411...

4 780.2794068067951456595241495989622...

5 13755.2719024115081712083954421541320...

6 296476.9162644200814909862281498491264...

7 7553550.6198338218721069097516499501996...

8 222082591.6017202421029000117685530884167...

9 7400694480.0494436216324852038000444393262...

10 275651917450.6709238286995776605620357737005...

---------------------------------------------------

d(k) is a root of polynomial:

---------------------------------------------------

k=2, 1 - 6*d + d^2

k=3, -1 + 3*d - 57*d^2 + d^3

k=4, 1 - 12*d - 218*d^2 - 780*d^3 + d^4

k=5, -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5

k=6, 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6

k=7, -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7

k=8, 1 - 24*d - 102692*d^2 - 9298344*d^3 + 536208070*d^4 - 7106080680*d^5 - 1688209700*d^6 - 222082584*d^7 + d^8

(End)

d(k) = (2^(1/k) - 1)^(-k). - David Bevan, Apr 07 2022

d(k) is asymptotic to (k/log(2))^k/sqrt(2). - David Bevan, Apr 07 2022

A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column. - Andrew Howroyd, Jan 23 2020

Examples

			A(2,2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].
Square array A(n,k) begins:
  1, 1,    1,        1,             1,                   1, ...
  1, 1,    3,       13,            75,                 541, ...
  1, 1,   13,      409,         23917,             2244361, ...
  1, 1,   63,    16081,      10681263,         14638956721, ...
  1, 1,  321,   699121,    5552351121,     117029959485121, ...
  1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..48, flattened
• J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Crossrefs

Columns: A000012 (k=0 and k=1), A001850 (k=2), A126086 (k=3), A263064 (k=4), A263065 (k=5), A263066 (k=6), A263067 (k=7), A263068 (k=8), A263069 (k=9), A263070 (k=10).

Rows: A000012 (n=0), A000670 (n=1), A055203 (n=2), A062208 (n=3), A062205 (n=4), A263061 (n=5), A263062 (n=6), A062204 (n=7), A263063 (n=8), A263071 (n=9), A263072 (n=10).

Main diagonal: A262810.

Cf. A210472, A225094, A227578, A227655, A229142, A229345, A263159, A316674.

Cf. A188392, A330942, A331461, A331637.

Programs

Maple

A:= (n, k)-> add(add((-1)^i*binomial(j, i)*
     binomial(j-i, n)^k, i=0..j), j=0..k*n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

A[, 0] =  1; A[n, k_] := Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jul 22 2016, after Alois P. Heinz *)

PARI

T(n,k) = {my(m=n*k); sum(j=0, m, binomial(j,n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020

Formula

A(n,k) = Sum_{j=0..k*n} Sum_{i=0..j} (-1)^i*C(j,i)*C(j-i,n)^k.

A(n,k) = Sum_{i >= 0} binomial(i,n)^k/2^(i+1). - Peter Bala, Jan 30 2018

A(n,k) = Sum_{j=0..n*k} binomial(j,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020