A337127 Table with 10 columns read by rows: T(n, k) is the number of n-digit positive integers with exactly k distinct base 10 digits (0 < k <= 10).

9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 81, 0, 0, 0, 0, 0, 0, 0, 0, 9, 243, 648, 0, 0, 0, 0, 0, 0, 0, 9, 567, 3888, 4536, 0, 0, 0, 0, 0, 0, 9, 1215, 16200, 45360, 27216, 0, 0, 0, 0, 0, 9, 2511, 58320, 294840, 408240, 136080, 0, 0, 0, 0, 9, 5103, 195048, 1587600, 3810240, 2857680, 544320, 0, 0, 0

Offset

1,1

Links

Table of n, a(n) for n=1..70.

Index entries for sequences related to digits.

Formula

T(n, k) = 9*Pochhammer(11-k, k-1)*n! * [x^n] (exp(x) - 1)^k/k!.

T(n, k) = 9*Pochhammer(11-k, k-1) * [x^n] x^k/Product_{j=1..k} (1-j*x).

T(n, k) = 9*Pochhammer(11-k, k-1)*S2(n, k) where S2(n, k) = A048993(n, k) are the Stirling numbers of the 2nd kind.

Example

The table T(n, k) begins:
9     0      0       0       0       0  0  0  0  0
9    81      0       0       0       0  0  0  0  0
9   243    648       0       0       0  0  0  0  0
9   567   3888    4536       0       0  0  0  0  0
9  1215  16200   45360   27216       0  0  0  0  0
9  2511  58320  294840  408240  136080  0  0  0  0
...

Mathematica

T[n_, k_]:=9Pochhammer[11-k, k-1]/k!*n!*Coefficient[Series[(Exp[x]-1)^k, {x, 0, n}], x, n]; Table[T[n, k], {n, 7}, {k, 10}]//Flatten

Crossrefs

Cf. A010785, A031955, A031962, A031969, A031987, A116670, A171102, A218019, A219743, A220076.

Cf. A010734, A048993, A052268 (row sums), A073531 (diagonal), A180599 (k = 1), A335843 (k = 2), A337313 (k = 3).

Sequence in context: A019440 A085276 A226121 * A193581 A270034 A228635

Adjacent sequences: A337124 A337125 A337126 * A337128 A337129 A337130

Keyword

nonn,tabf,base

Author

Stefano Spezia, Aug 17 2020

Status

approved