A123577 The Kruskal-Macaulay function L_5(n).

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 13, 13, 13, 13, 14, 14, 14, 15, 15, 16, 18, 18, 18, 19, 19, 20, 22, 22, 23, 25, 28, 28, 28, 28, 28, 29, 29, 29, 29, 30, 30, 30, 31, 31, 32, 34, 34, 34, 34, 35, 35, 35, 36, 36

Offset

0,12

Comments

Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then L_t(n) = C(n_t,t+1) + C(n_{t-1},t) + ... + C(n_v,v+1).

References

D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.

Links

Table of n, a(n) for n=0..79.

Maple

lowpol := proc(n, t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x, t) <= n do x := x+1 ; od ; RETURN(x-1) ; end: C := proc(n, t) local nresid, tresid, m, a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid, tresid) ; a := [op(a), m] ; nresid := nresid - binomial(m, tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end: L := proc(n, t) local a ; a := C(n, t) ; add( binomial(op(i, a), t+2-i), i=1..nops(a)) ; end: A123577 := proc(n) L(n, 5) ; end: for n from 0 to 80 do printf("%d, ", A123577(n)) ; od ; # R. J. Mathar, May 18 2007

Mathematica

(* The function L(n, t) is defined in A123575 *)
a[n_] := L[n, 5];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020 *)

Crossrefs

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.

Sequence in context: A120194 A120195 A121279 * A005854 A169991 A035435

Adjacent sequences: A123574 A123575 A123576 * A123578 A123579 A123580

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 12 2006

Extensions

More terms from R. J. Mathar, May 18 2007

Status

approved