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A123577 The Kruskal-Macaulay function L_5(n). 3

%I

%S 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,

%T 10,11,13,13,13,13,14,14,14,15,15,16,18,18,18,19,19,20,22,22,23,25,28,

%U 28,28,28,28,29,29,29,29,30,30,30,31,31,32,34,34,34,34,35,35,35,36,36

%N The Kruskal-Macaulay function L_5(n).

%C 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).

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

%p 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

%t (* The function L(n,t) is defined in A123575 *)

%t a[n_] := L[n, 5];

%t a /@ Range[0, 80] (* _Jean-Fran├žois Alcover_, Mar 29 2020 *)

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

%K nonn,easy

%O 0,12

%A _N. J. A. Sloane_, Nov 12 2006

%E More terms from _R. J. Mathar_, May 18 2007

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)