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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. 33
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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

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

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

EXAMPLE

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, ...

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-Fran├žois Alcover, Jul 22 2016, after Alois P. Heinz *)

PROG

(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

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.

Sequence in context: A348988 A257565 A276121 * A331568 A010278 A137795

Adjacent sequences:  A262806 A262807 A262808 * A262810 A262811 A262812

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Oct 02 2015

STATUS

approved

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Last modified May 22 04:12 EDT 2022. Contains 353933 sequences. (Running on oeis4.)