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A123579 The Kruskal-Macaulay function M_3(n). 4

%I

%S 0,1,2,3,3,4,5,5,6,6,6,7,8,8,9,9,9,10,10,10,10,11,12,12,13,13,13,14,

%T 14,14,14,15,15,15,15,15,16,17,17,18,18,18,19,19,19,19,20,20,20,20,20,

%U 21,21,21,21,21,21,22,23,23,24,24,24,25,25,25,25,26,26,26,26,26,27,27,27

%N The Kruskal-Macaulay function M_3(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 M_t(n) = C(n_t-1,t-1) + C(n_{t-1}-1,t-2) + ... + C(n_v-1,v-1).

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

%H B. M. Abrego, S. Fernandez-Merchant, B. Llano, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Abrego/abrego2.html">An Inequality for Macaulay Functions</a>, J. Int. Seq. 14 (2011) # 11.7.4

%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:

%p 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:

%p M := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a)-1,t-i),i=1..nops(a)) ; end:

%p A123579 := proc(n) M(n,3) ; end:

%p for n from 0 to 120 do printf("%d, ",A123579(n)) ; od ; # _R. J. Mathar_, Mar 14 2007

%Y For M_i(n), i=1, 2, 3, 4, 5 see A000127, A123578, A123579, A123580, A123731.

%K nonn,easy

%O 0,3

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

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Last modified August 13 07:13 EDT 2022. Contains 356078 sequences. (Running on oeis4.)