querying multidimensional sequences in OEIS

[Max A.]

On 11/3/06, Marc LeBrun <mlb at well.com> wrote:

> B. Encode each tuple as a single number, say by using it as factor
> exponents:  T(i) --> 2^T(i,1) * 3^T(i,2) * 5^T(i,3) *...

The sequences I've mentioned encoded this way:

I) Multinomial coefficients:

a( [k1, k2, ..., kn] ) = (k1 + k2 +  ... + kn)! / (k1! * k2! * ... * kn!)

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1,
4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2,
1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2,
6, 1, 10, 1, 2, 3, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2, 2, 4, 1, 12, 2,
3, 2, 2, 2, 6, 1, 3, 3, 6

II) Coefficients of incomplete Bell polynomials:

b( [k1, k2, ..., kn] ) = (k1 + 2*k2 + ... + n*kn)! / (k1! * k2! * ...
* kn!) / (1!^k1 * 2!^k2 * ... * n!^kn)

1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 1, 5, 10, 1, 1, 15, 1, 10, 15, 6,
1, 10, 10, 7, 15, 15, 1, 60, 1, 1, 21, 8, 35, 45, 1, 9, 28, 20, 1,
105, 1, 21, 105, 10, 1, 15, 35, 70, 36, 28, 1, 105, 56, 35, 45, 11, 1,
210, 1, 12, 210, 1, 84, 168, 1, 36, 55, 280, 1, 105, 1, 13, 280, 45,
126, 252, 1, 35, 105, 14, 1, 420, 120, 15, 66, 56, 1, 840, 210, 55,
78, 16, 165, 21, 1, 315, 378, 280

III) The number of permutations with given cycle structure:

c( [k1, k2, ..., kn] ) = (k1 + 2*k2 + ... + n*kn)! / (k1! * k2! * ...
* kn!) / (1^k1 * 2^k2 * ... * n^kn)

1, 1, 1, 1, 2, 3, 6, 1, 3, 8, 24, 6, 120, 30, 20, 1, 720, 15, 5040,
20, 90, 144, 40320, 10, 40, 840, 15, 90, 362880, 120, 3628800, 1, 504,
5760, 420, 45, 39916800, 45360, 3360, 40, 479001600, 630, 6227020800,
504, 210, 403200, 87178291200, 15, 1260, 280, 25920, 3360,
1307674368000, 105, 2688, 210, 226800, 3991680, 20922789888000, 420,
355687428096000, 43545600, 1260, 1, 20160, 4032, 6402373705728000,
25920, 2217600, 3360, 121645100408832000, 105, 2432902008176640000,
518918400, 1120, 226800, 18144, 30240, 51090942171709440000, 70, 105,
6706022400, 1124000727777607680000, 2520, 172800, 93405312000,
23950080, 1344, 25852016738884976640000, 1680, 151200, 2217600,
283046400, 1394852659200, 1663200, 21, 620448401733239439360000,
11340, 9072, 1120

Max