A010030 - OEIS

A010030

Irregular triangle read by rows: T(n,k) (n >= 1, 0 <= k <= [n/2]) = number of permutations of 1..n with [n/2]-k runs of consecutive pairs up and down (divided by 2).

1, 1, 0, 3, 0, 3, 8, 1, 25, 28, 7, 17, 155, 143, 45, 259, 1005, 933, 323, 131, 2770, 7488, 7150, 2621, 3177, 27978, 64164, 62310, 23811, 1281, 51433, 294602, 619986, 607445, 239653, 45155, 740375, 3331790, 6646750, 6545935, 2648395

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OFFSET

1,4

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.

LINKS

Seiichi Manyama, Rows n = 1..150, flattened

FORMULA

G.f. for number of permutations of 1..n by number of runs of consecutive pairs up and down is Sum_{n>=0} n! * ( ((1-y)*(2*x^2-x^3)-x)/((1-y)*x^2-1) )^n, cf. A010029. - Vladeta Jovovic, Nov 23 2007

EXAMPLE

Irregular triangle begins:

1, 0;

3, 0;

3, 8, 1;

25, 28, 7;

17, 155, 143, 45;

259, 1005, 933, 323;

131, 2770, 7488, 7150, 2621;

3177, 27978, 64164, 62310, 23811;

1281, 51433, 294602, 619986, 607445, 239653;

...

CROSSREFS

Row sums give A001710.

Cf. A002464, A001266, A000239, A000544, A001282, A398330, A398331, A398328.

Sequence in context: A021771 A154853 A139214 * A372339 A197270 A117940

Adjacent sequences: A010027 A010028 A010029 * A010031 A010032 A010033

KEYWORD

tabf,nonn,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Nov 23 2007

Entry revised by N. J. A. Sloane, Apr 14 2014

Triangle extended from row n=11 by Seiichi Manyama, Sep 01 2026.

STATUS

approved