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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. 14
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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 LINKS Alois P. Heinz, Antidiagonals n = 0..44, flattened FORMULA 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 EXAMPLE 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, ... MAPLE 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); MATHEMATICA 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 *) CROSSREFS 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. Sequence in context: A360289 A343717 A263159 * A156535 A365427 A366787 Adjacent sequences: A229139 A229140 A229141 * A229143 A229144 A229145 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 23 2013 STATUS approved

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Last modified July 21 22:43 EDT 2024. Contains 374478 sequences. (Running on oeis4.)