This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A064044 Square array read by antidiagonals of number of length k walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part. 4
 1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 18, 12, 4, 1, 0, 10, 60, 51, 20, 5, 1, 0, 20, 200, 234, 108, 30, 6, 1, 0, 35, 700, 1110, 624, 195, 42, 7, 1, 0, 70, 2450, 5460, 3760, 1350, 318, 56, 8, 1, 0, 126, 8820, 27405, 23480, 9770, 2556, 483, 72, 9, 1, 0, 252 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - Paul D. Hanna, Apr 07 2005 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 FORMULA a(n,k) = Sum{j=0..k} C(k,j) B(j) a(n-1,k-j) where B(j) = C(j,[j/2]) = A001405(j) with a(0,0) = 1 and a(0,k) = 0 for k>0. E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - Paul D. Hanna, Apr 07 2005 EXAMPLE Rows start: 1, 0,  0,   0,    0,     0,      0, ... 1, 1,  2,   3,    6,    10,     20, ... 1, 2,  6,  18,   60,   200,    700, ... 1, 3, 12,  51,  234,  1110,   5460, ... 1, 4, 20, 108,  624,  3760,  23480, ... 1, 5, 30, 195, 1350,  9770,  73300, ... 1, 6, 42, 318, 2556, 21480, 187140, ... MAPLE a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),        add(binomial(k, j)*binomial(j, floor(j/2))        *a(n-1, k-j), j=0..k))     end: seq(seq(a(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 06 2014 MATHEMATICA a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[k, j]*Binomial[j, Floor[j/2]]*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *) PROG (PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); k!*polcoeff(polcoeff(1/(1-X*besseli(0, 2*Y)-X*Y*besseli(1, 2*Y)), n, x), k, y)} /* Hanna */ CROSSREFS Rows include A000007, A001405, A005566, A064036. Columns include A000012, A001477, A002378, A064043. Cf. A064045. Sequence in context: A185937 A292086 A065177 * A213980 A144912 A306708 Adjacent sequences:  A064041 A064042 A064043 * A064045 A064046 A064047 KEYWORD nonn,tabl AUTHOR Henry Bottomley, Aug 23 2001 STATUS approved

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

Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)