cp's OEIS Frontend

A123580 The Kruskal-Macaulay function M_4(n).

%I A123580 #17 Mar 30 2023 09:14:49
%S A123580 0,1,2,3,4,4,5,6,7,7,8,9,9,10,10,10,11,12,13,13,14,15,15,16,16,16,17,
%T A123580 18,18,19,19,19,20,20,20,20,21,22,23,23,24,25,25,26,26,26,27,28,28,29,
%U A123580 29,29,30,30,30,30,31,32,32,33,33,33,34,34,34,34,35,35,35,35,35,36,37,38
%N A123580 The Kruskal-Macaulay function M_4(n).
%C A123580 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 A123580 D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.
%H A123580 B. M. Abrego, S. Fernandez-Merchant, and 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 A123580 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: M := proc(n,t) local a; a := C(n,t); add( binomial(op(i,a)-1,t-i),i=1..nops(a)); end: A123580 := proc(n) M(n,4); end: for n from 0 to 120 do printf("%d, ",A123580(n)); od; # _R. J. Mathar_, Mar 14 2007
%t A123580 lowpol[n_, t_] := Module[{x = Floor[(n*t!)^(1/t)]}, While[Binomial[x, t] <= n, x = x + 1]; x - 1];
%t A123580 c[n_, t_] := Module[{n0 = n, t0 = t, a = {}, m}, While[n0 > 0, m = lowpol[n0, t0]; AppendTo[a, m]; n0 = n0 - Binomial[m, t0]; t0 = t0 - 1]; a];
%t A123580 M[n_, t_] := With[{a = c[n, t]}, Sum[Binomial[a[[i]] - 1, t - i], {i, 1, Length[a]}]];
%t A123580 A123580[n_] := M[n, 4];
%t A123580 Table[A123580[n], {n, 0, 73}] (* _Jean-François Alcover_, Mar 30 2023, after _R. J. Mathar_ *)
%Y A123580 For M_i(n), i=1, 2, 3, 4, 5 see A000127, A123578, A123579, A123580, A123731.
%K A123580 nonn,easy
%O A123580 0,3
%A A123580 _N. J. A. Sloane_, Nov 12 2006