This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A262809 #58 Apr 09 2022 09:48:58
%S A262809 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,
%T A262809 321,1,1,1,4683,2244361,10681263,699121,1683,1,1,1,47293,308682013,
%U A262809 14638956721,5552351121,32193253,8989,1,1,1,545835,58514835289,35941784497263,117029959485121,3147728203035,1538743249,48639,1,1
%N 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.
%C A262809 Also, A(n,k) is the number of alignments for k sequences of length n each (Slowinski 1998).
%C A262809 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
%C A262809 From _Vaclav Kotesovec_, Mar 23 2016: (Start)
%C A262809 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.
%C A262809 ---------------------------------------------------
%C A262809 k d(k)
%C A262809 ---------------------------------------------------
%C A262809 2 5.8284271247461900976033774484193...
%C A262809 3 56.9476283720414911685286267804411...
%C A262809 4 780.2794068067951456595241495989622...
%C A262809 5 13755.2719024115081712083954421541320...
%C A262809 6 296476.9162644200814909862281498491264...
%C A262809 7 7553550.6198338218721069097516499501996...
%C A262809 8 222082591.6017202421029000117685530884167...
%C A262809 9 7400694480.0494436216324852038000444393262...
%C A262809 10 275651917450.6709238286995776605620357737005...
%C A262809 ---------------------------------------------------
%C A262809 d(k) is a root of polynomial:
%C A262809 ---------------------------------------------------
%C A262809 k=2, 1 - 6*d + d^2
%C A262809 k=3, -1 + 3*d - 57*d^2 + d^3
%C A262809 k=4, 1 - 12*d - 218*d^2 - 780*d^3 + d^4
%C A262809 k=5, -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5
%C A262809 k=6, 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6
%C A262809 k=7, -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7
%C A262809 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
%C A262809 (End)
%C A262809 d(k) = (2^(1/k) - 1)^(-k). - _David Bevan_, Apr 07 2022
%C A262809 d(k) is asymptotic to (k/log(2))^k/sqrt(2). - _David Bevan_, Apr 07 2022
%C A262809 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
%H A262809 Alois P. Heinz, <a href="/A262809/b262809.txt">Antidiagonals n = 0..48, flattened</a>
%H A262809 J. B. Slowinski, <a href="http://www.neurociencias.org.ve/cont-cursos-laboratorio-de-neurociencias-luz/Slowinski1998%20phylogenetics.pdf">The Number of Multiple Alignments</a>, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:<a href="http://dx.doi.org/10.1006/mpev.1998.0522">10.1006/mpev.1998.0522</a>
%F A262809 A(n,k) = Sum_{j=0..k*n} Sum_{i=0..j} (-1)^i*C(j,i)*C(j-i,n)^k.
%F A262809 A(n,k) = Sum_{i >= 0} binomial(i,n)^k/2^(i+1). - _Peter Bala_, Jan 30 2018
%F A262809 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
%e A262809 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)].
%e A262809 Square array A(n,k) begins:
%e A262809 1, 1, 1, 1, 1, 1, ...
%e A262809 1, 1, 3, 13, 75, 541, ...
%e A262809 1, 1, 13, 409, 23917, 2244361, ...
%e A262809 1, 1, 63, 16081, 10681263, 14638956721, ...
%e A262809 1, 1, 321, 699121, 5552351121, 117029959485121, ...
%e A262809 1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, ...
%p A262809 A:= (n, k)-> add(add((-1)^i*binomial(j, i)*
%p A262809 binomial(j-i, n)^k, i=0..j), j=0..k*n):
%p A262809 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A262809 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}];
%t A262809 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Jul 22 2016, after _Alois P. Heinz_ *)
%o A262809 (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
%Y A262809 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).
%Y A262809 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).
%Y A262809 Main diagonal: A262810.
%Y A262809 Cf. A210472, A225094, A227578, A227655, A229142, A229345, A263159, A316674.
%Y A262809 Cf. A188392, A330942, A331461, A331637.
%K A262809 nonn,tabl
%O A262809 0,8
%A A262809 _Alois P. Heinz_, Oct 02 2015