0,8

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

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

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

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, ...

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);

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

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.

Sequence in context: A348988 A257565 A276121 * A331568 A010278 A137795

Adjacent sequences: A262806 A262807 A262808 * A262810 A262811 A262812

nonn,tabl

Alois P. Heinz, Oct 02 2015

approved