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 A123575 The Kruskal-Macaulay function L_3(n). 3
 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 5, 5, 5, 6, 6, 7, 9, 9, 10, 12, 15, 15, 15, 16, 16, 17, 19, 19, 20, 22, 25, 25, 26, 28, 31, 35, 35, 35, 36, 36, 37, 39, 39, 40, 42, 45, 45, 46, 48, 51, 55, 55, 56, 58, 61, 65, 70, 70, 70, 71, 71, 72, 74, 74, 75, 77, 80, 80, 81, 83, 86, 90, 90, 91, 93 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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). REFERENCES D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3. LINKS MAPLE 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+i), i=1..nops(a)) ; add( binomial(op(i, a), t+2-i), i=1..nops(a)) ; end: A123575 := proc(n) L(n, 3) ; end: for n from 0 to 80 do printf("%d, ", A123575(n)) ; od ; # R. J. Mathar, May 18 2007 MATHEMATICA lowpol[n_, t_] := Module[{x = Floor[(n t!)^(1/t)]}, While[Binomial[x, t] <= n, x++] ; x - 1]; c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; AppendTo[a, m]; n0 -= Binomial[m, t0]; t0--]; a]; L[n_, t_] := Module[{a = c[n, t]}, Sum[Binomial[a[[i]], t + 2 - i], {i, 1, Length[a]}]]; a[n_] := L[n, 3]; a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020, after R. J. Mathar *) CROSSREFS For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577. Sequence in context: A115126 A055769 A162217 * A014200 A293522 A319476 Adjacent sequences:  A123572 A123573 A123574 * A123576 A123577 A123578 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Nov 12 2006 EXTENSIONS More terms from R. J. Mathar, May 18 2007 STATUS approved

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Last modified August 13 08:27 EDT 2020. Contains 336442 sequences. (Running on oeis4.)