A164933 Word structures of length n using a 10-ary alphabet, including a special character, which may occur in the leftmost word position only if n=1.

1, 2, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562594, 3534961, 23428397, 163187870, 1190538144, 9066908419, 71837342107, 590009213152, 5004416730974, 43667740987637, 390497323261737, 3565602957116674, 33130941174471484, 312322981161532615, 2979191694795132887

Offset

0,2

Comments

a(n) is also the number of ways of placing n labeled balls into 9 indistinguishable boxes and one special box, where the first ball is allowed to be member of the special box only if n=1.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{m=0..n-1} C(n-1,m) Sum_{k=0..9} S2 (n-m, k), if n>1; a(n) = n+1 else.

a(n) = 2119/11520*2^n +103/840*3^n +53/1152*4^n +11/900*5^n +6^n/384 +7^n/2520 +8^n/11520 +10^n/403200, if n>1; a(n) = n+1 else.

G.f.: (403200*x^9 -478089*x^8 +35157*x^7 +202072*x^6 -136061*x^5 +42574*x^4 -7538*x^3 +774*x^2 -43*x+1) / ((2*x-1)* (3*x-1)* (4*x-1)* (5*x-1)* (6*x-1)* (7*x-1)* (8*x-1)* (10*x-1)).

Example

a(0) = 1, the only possible word structure is the empty word.
a(1) = 2, word structures are a and X, where X denotes the special character.
a(2) = 3, word structures are aa, ab, aX.
a(3) = 10, word structures are aaa, aab, aba, baa, abc, aaX, abX, aXa, aXb, aXX.

Maple

# first program:
a:= n-> `if`(n<2, n+1, 2119/11520*2^n +103/840*3^n +53/1152*4^n +11/900*5^n +6^n/384 +7^n/2520 +8^n/11520 +10^n/403200): seq(a(n), n=0..30);
# second program:
a:= n-> `if`(n<2, n+1, add(add(Stirling2(n-m, k), k=0..9) *binomial(n-1, m), m=0..n-1)): seq(a(n), n=0..30);

Crossrefs

Cf. A007318, A048993, A164863, A164864, A000079, A000244, A007582, A086444, A005493.

Sequence in context: A296269 A141102 A144720 * A003048 A008980 A064183

Adjacent sequences: A164930 A164931 A164932 * A164934 A164935 A164936

Keyword

easy,nonn

Author

Alois P. Heinz, Aug 31 2009

Status

approved