The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
 1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 20, 4, 0, 1, 8, 54, 176, 10, 0, 1, 10, 104, 996, 1876, 28, 0, 1, 12, 170, 2944, 22734, 22064, 84, 0, 1, 14, 252, 6500, 108136, 577692, 275568, 264, 0, 1, 16, 350, 12144, 332050, 4525888, 15680628, 3584064, 858, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Column k is INVERTi transform of k-th row of A287318. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,1)/2 = A000108(n-1) for n >= 1. G.f. of column k: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^k dt. - Shel Kaphan, Mar 19 2023 EXAMPLE Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 2, 4, 6, 8, 10, 12, ... 0, 2, 20, 54, 104, 170, 252, ... 0, 4, 176, 996, 2944, 6500, 12144, ... 0, 10, 1876, 22734, 108136, 332050, 796860, ... 0, 28, 22064, 577692, 4525888, 19784060, 62039088, ... MAPLE b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(b(n-j, i-1)*binomial(n, j)^2, j=0..n)) end: g:= proc(n, k) option remember; `if` (n<1, -1, -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n)) end: A:= (n, k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))): seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1; b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *) g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *) a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}]; TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* Shel Kaphan, Mar 14 2023 *) CROSSREFS Columns k=0-5 give: A000007, |A002420|, A054474, A049037, A359801, A361364. Rows n=0-2 give: A000012, A005843, A139271. Main diagonal gives A361297. Cf. A000108, A287318. Sequence in context: A302996 A266213 A289522 * A316273 A124915 A322084 Adjacent sequences: A361394 A361395 A361396 * A361398 A361399 A361400 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 10 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 15 12:56 EDT 2024. Contains 374332 sequences. (Running on oeis4.)