A092285 Number of productions of a certain "divide-and-conquer" context-free grammar in Chomsky normal form that generates all permutations of n symbols.

1, 4, 12, 22, 65, 116, 399, 554, 2475, 3232, 14938, 20208, 101413, 130846, 691890, 924946, 4867559, 6281552, 35154066, 46902128, 253485141, 328375720, 1877693458, 2504042868, 13754442225, 17885555976, 103317302754, 137761862662, 765705075439, 998817493980

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

P. R. J. Asveld, Generating all permutations by context-free grammars in Chomsky normal form, Theoretical Computer Science 354 (2006) 118-130.

Formula

a(n) = Sum_{k=1..n} t(n, k)*C(k, ceiling(k/2)), where t(n, k) is the n-th row in the Pascal-like triangle of A090349 and C(k, i) is the binomial coefficient.

Example

a(4) = 4*C(1,1) + 6*C(2,1) + 0*C(3,2) + 1*C(4,2) = 4 + 12 + 0 + 6 = 22; cf. the example grammar in A090349.

Prog

(PARI) a(n)={my(b=vector(n)); for(i=0, logint(n, 2), b[n>>i]=1; b[((n-1)>>i)+1]=1); sum(k=1, n, b[k]*binomial(n, k)*binomial(k, k\2))} \\ Andrew Howroyd, Feb 29 2020

Crossrefs

Cf. A090349.

Sequence in context: A008161 A008243 A008194 * A016440 A301191 A008010

Adjacent sequences: A092282 A092283 A092284 * A092286 A092287 A092288

Keyword

nonn

Author

Peter R. J. Asveld, Jan 30 2004

Extensions

Terms a(21) and beyond from Andrew Howroyd, Feb 29 2020

Status

approved