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A123731 The Kruskal-Macaulay function M_5(n).

%I A123731 #14 Mar 30 2023 09:14:32
%S A123731 0,1,2,3,4,5,5,6,7,8,9,9,10,11,12,12,13,14,14,15,15,15,16,17,18,19,19,
%T A123731 20,21,22,22,23,24,24,25,25,25,26,27,28,28,29,30,30,31,31,31,32,33,33,
%U A123731 34,34,34,35,35,35,35,36,37,38,39,39,40,41,42,42,43,44,44,45,45,45,46
%N A123731 The Kruskal-Macaulay function M_5(n).
%C A123731 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 A123731 D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.
%p A123731 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 A123731 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 A123731 M := proc(n,t) local a; a := C(n,t); add( binomial(op(i,a)-1,t-i),i=1..nops(a)); end:
%p A123731 A123731 := proc(n) M(n,5); end:
%p A123731 for n from 0 to 120 do printf("%d, ",A123731(n)); od; # _R. J. Mathar_, Mar 14 2007
%t A123731 lowpol[n_, t_] := Module[{x = Floor[(n*t!)^(1/t)]}, While[Binomial[x, t] <= n, x = x + 1]; x - 1];
%t A123731 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 A123731 M[n_, t_] := With[{a = c[n, t]}, Sum[Binomial[a[[i]] - 1, t - i], {i, 1, Length[a]}]];
%t A123731 A123731[n_] := M[n, 5];
%t A123731 Table[A123731[n], {n, 0, 72}] (* _Jean-François Alcover_, Mar 30 2023, after _R. J. Mathar_ *)
%Y A123731 For M_i(n), i=1, 2, 3, 4, 5 see A000127, A123578, A123579, A123580, A123731.
%K A123731 nonn,easy
%O A123731 0,3
%A A123731 _N. J. A. Sloane_, Nov 12 2006
%E A123731 More terms from _R. J. Mathar_, Mar 14 2007