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A123573 The Kruskal-Macaulay function K_4(n).

%I A123573 #13 Mar 30 2023 09:14:37
%S A123573 0,4,7,9,10,10,13,15,16,16,18,19,19,20,20,20,23,25,26,26,28,29,29,30,
%T A123573 30,30,32,33,33,34,34,34,35,35,35,35,38,40,41,41,43,44,44,45,45,45,47,
%U A123573 48,48,49,49,49,50,50,50,50,52,53,53,54,54,54,55,55,55,55,56,56,56,56,56
%N A123573 The Kruskal-Macaulay function K_4(n).
%C A123573 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 K_t(n) = C(n_t,t-1) + C(n_{t-1},t-2) + ... + C(n_v,v-1).
%D A123573 D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.
%p A123573 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: K := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a),t-i),i=1..nops(a)) ; end: A123573 := proc(n) K(n,4) ; end: for n from 0 to 80 do printf("%d, ",A123573(n)) ; od ; # _R. J. Mathar_, May 18 2007
%t A123573 lowpol[n_, t_] := Module[{x}, x = Floor[(n*t!)^(1/t)]; While[Binomial[x, t] <= n, x = x + 1]; x - 1];
%t A123573 c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; a = Append[a, m]; n0 = n0 - Binomial[m, t0]; t0 = t0 - 1]; a];
%t A123573 K[n_, t_] := Module[{a}, a = c[n, t]; Sum[Binomial[a[[i]], t - i], {i, 1, Length[a]}]];
%t A123573 A123573[n_] := K[n, 4];
%t A123573 Table[A123573[n], {n, 0, 70}] (* _Jean-François Alcover_, Mar 30 2023, after _R. J. Mathar_ *)
%Y A123573 For K_i(n), i=1, 2, 3, 4, 5 see A000012, A003057, A123572, A123573, A123574.
%K A123573 nonn,easy
%O A123573 0,2
%A A123573 _N. J. A. Sloane_, Nov 12 2006
%E A123573 More terms from _R. J. Mathar_, May 18 2007