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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-François 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 December 2 09:50 EST 2023. Contains 367517 sequences. (Running on oeis4.)