A227655 -id:A227655 - cp's OEIS frontend

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

Alois P. Heinz, Oct 02 2015

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

			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, ...

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

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.

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 *)

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

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

A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 15, 157, 63, 1, 1, 1, 31, 2101, 5419, 321, 1, 1, 1, 63, 32461, 717795, 220561, 1683, 1, 1, 1, 127, 580693, 142090291, 328504401, 9763807, 8989, 1, 1, 1, 255, 11917837, 39991899123, 944362553521, 172924236255, 454635973, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 11 2015

			Square array A(n,k) begins:
  1, 1,    1,       1,            1,                1, ...
  1, 1,    3,       7,           15,               31, ...
  1, 1,   13,     157,         2101,            32461, ...
  1, 1,   63,    5419,       717795,        142090291, ...
  1, 1,  321,  220561,    328504401,     944362553521, ...
  1, 1, 1683, 9763807, 172924236255, 7622403922836151, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A115866, A263162, A263163, A263164, A263165, A263166, A263167, A263168.
Rows n=0-1 give: A000012, A255047.
Main diagonal gives A263160.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A262809.

s:= proc(n) option remember; `if`(n=0, {[]},
      map(x-> [[x[], 0], [x[], 1]][], s(n-1)))
    end:
b:= proc(l) option remember; `if`(l=[] or l[1]=0, 1,
       add((p-> `if`(p[1]<0, 0, `if`(p[1]=0, 1, b(p)))
       )(sort(l-x)), x=s(nops(l)) minus {[0$nops(l)]}))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n,d-n), n=0..d), d=0..10);

g[k_] := Table[Reverse[IntegerDigits[n, 2]][[;;k]], {n, 2^k+1, 2^(k+1)-1}];
b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[k]}]];
a[n_, k_] := If[n == 0 || k == 0 || k == 1, 1, b[Table[n, {k}]]];
Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz in A115866 *)

A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 25, 115, 63, 1, 1, 1, 121, 2641, 2371, 321, 1, 1, 1, 721, 114121, 392641, 54091, 1683, 1, 1, 1, 5041, 7489441, 169417921, 67982041, 1307377, 8989, 1, 1, 1, 40321, 681120721, 137322405361, 308238414121, 12838867105, 32803219, 48639, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2013

Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j - Product_{j=1..k} x_j) for k>1. - Seiichi Manyama, Jul 10 2020

			A(1,3) = 3*2+1 = 7:
          (0,1,1)-(0,0,1)
         /       X       \
  (1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
       \ \       X       / /
        \ (1,1,0)-(1,0,0) /
         `---------------´
Square array A(n,k) begins:
  1, 1,    1,       1,           1,               1, ...
  1, 1,    3,       7,          25,             121, ...
  1, 1,   13,     115,        2641,          114121, ...
  1, 1,   63,    2371,      392641,       169417921, ...
  1, 1,  321,   54091,    67982041,    308238414121, ...
  1, 1, 1683, 1307377, 12838867105, 629799991355641, ...

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A081798, A082488, A082489, A229049, A229674, A229675, A229676, A229677.
Rows n=0-1 give: A000012, A038507 (for k>1).
Main diagonal gives: A229267.
Cf. A060854, A227578, A227655, A225094, A210472, A262809, A263159.

with(combinat):
A:= (n,k)-> `if`(k<2, 1, add(multinomial(n+(k-1)*j, n-j, j$k), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

a[, 0] = a[, 1] = 1; a[n_, k_] := Sum[Product[Binomial[n+j*m, m], {j, 0, k-1}], {m, 0, n}]; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = Sum_{j=0..n} multinomial(n+(k-1)*j; n-j, {j}^k) for k>1, A(n,0) = A(n,1) = 1.

G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1-x)^(k*j+1). for k>1. - Seiichi Manyama, Jul 10 2020

A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 22, 4, 1, 1, 25, 248, 188, 8, 1, 1, 121, 6506, 11380, 1712, 16, 1, 1, 721, 292442, 2359348, 577124, 16098, 32, 1, 1, 5041, 19450082, 1088626684, 991365512, 30970588, 154352, 64, 1

Offset: 0

Alois P. Heinz, Sep 24 2013

			A(2,2) = 22: [(2,2),(1,1),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)], [(2,2),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(1,2),(1,0),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(0,2),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(0,1),(0,0)], [(2,2),(2,1),(2,0),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,0)], [(2,2),(2,0),(1,0),(0,0)], [(2,2),(2,0),(0,0)].
Square array A(n,k) begins:
  1,  1,     1,        1,            1,                 1, ...
  1,  1,     3,        7,           25,               121, ...
  1,  2,    22,      248,         6506,            292442, ...
  1,  4,   188,    11380,      2359348,        1088626684, ...
  1,  8,  1712,   577124,    991365512,     4943064622568, ...
  1, 16, 16098, 30970588, 453530591824, 25162900228200976, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-3 give: A000012, A011782, A132595(n+1), A229482.
Rows n=0-2 give: A000012, A038507 (for k>1), A229465.
Main diagonal gives: A229346.
Cf. A060854, A227578, A227655, A225094, A210472, A229142, A262809, A263159.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1,
      `if`(m>1, add(b(l-[j$m]), j=1..l[1]), 0)+
      add(add(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..m))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 24 2013

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}],  0] + Sum[Sum[b[Sort[ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]]; a[n_, k_] := b[Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A318191 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point p we have abs(p_{i}-p_{(i mod k)+1}) <= 1 and the first component used is p_1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 12, 4, 1, 1, 1, 24, 180, 72, 8, 1, 1, 1, 120, 4680, 5400, 432, 16, 1, 1, 1, 720, 187200, 914400, 162000, 2592, 32, 1, 1, 1, 5040, 10634400, 296438400, 178660800, 4860000, 15552, 64, 1, 1, 1, 40320, 813664800, 162273628800, 469551168000, 34907788800, 145800000, 93312, 128, 1, 1

Offset: 0

Alois P. Heinz, Jan 07 2019

			A(2,2) = 2^2 = 4:
                    (0,1)
                   /     \
  (2,2)-(1,2)-(1,1)       (0,0)
                   \     /
                    (1,0)
Square array A(n,k) begins:
  1, 1,  1,     1,         1,             1,                   1, ...
  1, 1,  1,     2,         6,            24,                 120, ...
  1, 1,  2,    12,       180,          4680,              187200, ...
  1, 1,  4,    72,      5400,        914400,           296438400, ...
  1, 1,  8,   432,    162000,     178660800,        469551168000, ...
  1, 1, 16,  2592,   4860000,   34907788800,     743761386086400, ...
  1, 1, 32, 15552, 145800000, 6820487308800, 1178106009360998400, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0+1, 2 give: A000012, A011782.
Rows n=0-2 give: A000012, A000142(n-1) for n>0, A322782/n for n>0.
Main diagonal gives A320443.
Cf. A227655.

b:= proc(l) option remember; (n-> `if`(n<2 or max(l[])=0, 1,
      add(`if`(l[i]=0 or 1 `if`(k<2 or n=0, 1, b([n-1, n$k-1])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[l_] := b[l] = With[{n = Length[l]}, If[n < 2 || Max[l ] == 0, 1, Sum[If[ l[[i]] == 0 ||1 < Abs[l[[If[i == 1, 0, i] - 1]] - l[[i]] + 1] || 1 < Abs[l[[If[i == n, 0, i] + 1]] - l[[i]] + 1], 0, b[ReplacePart[l, i -> l[[i]] - 1]]], {i, n}]]];
A[n_, k_] :=  If[k < 2 || n == 0, 1, b[Join[{n - 1}, Table[n, {k - 1}]]]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 13 2020, after Maple *)

A227656 Number of lattice paths from {2}^n to {0}^n using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_n) we have abs(p_{i}-p_{i+1}) <= 1.

Original entry on oeis.org

1, 1, 4, 44, 896, 29392, 1413792, 93770800, 8201380224, 914570667792, 126651310675680, 21323599202141616, 4289517397262212416, 1016086393608958657680, 279937626985917460931616, 88754294249179769383418160, 32085579878185717054048193280, 13119328150439580260369558815248

Offset: 0

Alois P. Heinz, Jul 19 2013

nonn

Number of linear extensions of garland or double fence poset. - Alexander Shashkov, Jul 26 2020

			a(2) = 2^2 = 4:
.
        (1,2)       (0,1)
       /     \     /     \
  (2,2)       (1,1)       (0,0)
       \     /     \     /
        (2,1)       (1,0)
.
a(3) = 44:
.
          (1,2,2)-(1,1,2)-(0,1,2)-(0,1,1)-(0,0,1)
         /       X       \       /       X       \
  (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
         \       X       /       \       X       /
          (2,2,1)-(2,1,1)-(2,1,0)-(1,1,0)-(1,0,0)

Alexander Shashkov, Table of n, a(n) for n = 0..227 (terms 0..23 from Alois P. Heinz)
Oscar J. Borenstein and Alexander Shashkov, Garland Recurrences, arXiv:1909.04215 [math.CO], 2019.
Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.

Row n=2 of A227655.

Cf. A000079.

a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.197278552664313325820060688708960349... and c = 4.4668518532326348084863454883501... - Vaclav Kotesovec, Dec 25 2018

A227665 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component by 1 such that for each point (p_1,p_2,p_3) we have abs(p_{i}-p_{i+1}) <= 1.

Original entry on oeis.org

1, 6, 44, 320, 2328, 16936, 123208, 896328, 6520712, 47437640, 345104904, 2510609608, 18264477064, 132872558664, 966636864776, 7032203170760, 51158695924872, 372175277815624, 2707544336559112, 19697160911545032, 143295215053933448, 1042460827200624200

Offset: 0

Alois P. Heinz, Jul 19 2013

			a(1) = 3! = 3*2*1 = 6:
            (0,1,1) - (0,0,1)
          /         X         \
  (1,1,1) - (1,0,1)   (0,1,0) - (0,0,0)
          \         X         /
            (1,1,0) - (1,0,0)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,2).

Column k=3 of A227655.

Cf. A000142.

a:= n-> (<<0|1>, <2|7>>^n. <<1, 6>>)[1, 1]:
seq(a(n), n=0..25);

G.f.: (x-1)/(2*x^2+7*x-1).

a(n) = 7*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(2)=6.

A227666 Number of lattice paths from {n}^4 to {0}^4 using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_4) we have abs(p_{i}-p_{i+1}) <= 1.

Original entry on oeis.org

1, 24, 896, 33904, 1281696, 48447504, 1831288096, 69221669104, 2616540574496, 98903777810704, 3738507768500896, 141313513441272304, 5341572177372667296, 201908456107703653904, 7632027293479058673696, 288486385024598708555504, 10904624832208006924120096

Offset: 0

Alois P. Heinz, Jul 19 2013

			a(1) = 4! = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (40,-89,220).

Column k=4 of A227655.

Cf. A000142.

a:= n-> ceil((<<0|1|0>, <0|0|1>, <220|-89|40>>^n.
        <<10/11, 24, 896>>)[1, 1]):
seq(a(n), n=0..25);

G.f.: (20*x^3-25*x^2+16*x-1)/(220*x^3-89*x^2+40*x-1).

a(n) = 40*a(n-1) -89*a(n-2) +220*a(n-3) for n>3, a(0)=1, a(1)=24, a(2)=896, a(3)=33904.

A227667 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_5) we have abs(p_{i}-p_{i+1}) <= 1.

Original entry on oeis.org

1, 120, 29392, 7453320, 1897242448, 482913033152, 122911984813568, 31283451053916800, 7962224756951452544, 2026535155335964884480, 515791104488454210243072, 131278484324109833244067840, 33412829924638979294019463168, 8504190228674549912505288509440

Offset: 0

Alois P. Heinz, Jul 19 2013

			a(1) = 5! = 120.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=5 of A227655.

Cf. A000142.

a:= n-> coeff(series((173568*x^8 -3773248*x^7 +10330944*x^6 -719888*x^5 +1468896*x^4 -35208*x^3 -3608*x^2 +170*x-1) / (-98304*x^9 +4024832*x^8 -36900032*x^7 +37771968*x^6 -3950640*x^5 +5084576*x^4 -23648*x^3 -9016*x^2 +290*x-1), x, n+1), x, n): seq(a(n), n=0..20);

G.f.: (173568*x^8 -3773248*x^7 +10330944*x^6 -719888*x^5 +1468896*x^4 -35208*x^3 -3608*x^2 +170*x-1) / (-98304*x^9 +4024832*x^8 -36900032*x^7 +37771968*x^6 -3950640*x^5 +5084576*x^4 -23648*x^3 -9016*x^2 +290*x-1).

A227668 Number of lattice paths from {n}^6 to {0}^6 using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_6) we have abs(p_{i}-p_{i+1}) <= 1.

Original entry on oeis.org

1, 720, 1413792, 2940381648, 6173789662504, 12981179566917088, 27297846037161958056, 57403822541579269311072, 120712076511505386344017520, 253839841305922000782983605664, 533787802709908480895773030991840, 1122477022599575074944649300433060288

Offset: 0

Alois P. Heinz, Jul 19 2013

			a(1) = 6! = 720.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=6 of A227655.

Cf. A000142.

G.f.: -(67322341490688000*x^20 +27865599006421811200*x^19 -8662296440668699099136*x^18 -16874516373657444974592*x^17 +27085359075023950995456*x^16 -1808862947855651445760*x^15 -1414381090428803492096*x^14 -3613053959743748878592*x^13
+1363837434430612756288*x^12 +93587353840530417152*x^11 -39568432789577322400*x^10 -4418148372274485344*x^9 +81199402070343168*x^8 +9376560118889840*x^7 -221663818632940*x^6 +2817001053384*x^5 -695598308*x^4 +124162308*x^3 -958790*x^2 +2064*x -1)
/ ( -13254524928000000*x^21 -271933737533440000*x^20 -5733118008692572160*x^19 +116739247952003395584000*x^18 +224369280219612051439616*x^17 -356501961858517247606784*x^16 +60501369697833888177152*x^15 +16382732593079984754944*x^14
+13383960686857306419456*x^13 -5739755254392736710336*x^12 -404736183389439184896*x^11 +144333130922005891104*x^10 +16946456787615943968*x^9 -22388465914256448*x^8 -24650905650633712*x^7 +544278444686228*x^6 -5641861520584*x^5 -1907901380*x^4 -244171188*x^3 +1549478*x^2 -2784*x +1).

cp's OEIS Frontend

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.

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A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A318191 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point p we have abs(p_{i}-p_{(i mod k)+1}) <= 1 and the first component used is p_1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A227656 Number of lattice paths from {2}^n to {0}^n using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_n) we have abs(p_{i}-p_{i+1}) <= 1.

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A227665 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component by 1 such that for each point (p_1,p_2,p_3) we have abs(p_{i}-p_{i+1}) <= 1.

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A227666 Number of lattice paths from {n}^4 to {0}^4 using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_4) we have abs(p_{i}-p_{i+1}) <= 1.

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