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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-Fran├žois 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

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