A123575 -id:A123575 - cp's OEIS frontend

A123576 The Kruskal-Macaulay function L_4(n).

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 11, 11, 11, 12, 12, 13, 15, 15, 16, 18, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 26, 26, 26, 27, 27, 28, 30, 30, 31, 33, 36, 36, 36, 37, 37, 38, 40, 40, 41, 43, 46, 46, 47, 49, 52, 56, 56, 56, 56, 57, 57, 57, 58

Offset: 0

N. J. A. Sloane, Nov 12 2006

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).

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

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.

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+2-i),i=1..nops(a)) ; end: A123576 := proc(n) L(n,4) ; end: for n from 0 to 80 do printf("%d, ",A123576(n)) ; od ; # R. J. Mathar, May 18 2007

(+ The function L(n,t) is defined in A123575 *)
a[n_] := L[n, 4];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020 *)

More terms from R. J. Mathar, May 18 2007

A123577 The Kruskal-Macaulay function L_5(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 13, 13, 13, 13, 14, 14, 14, 15, 15, 16, 18, 18, 18, 19, 19, 20, 22, 22, 23, 25, 28, 28, 28, 28, 28, 29, 29, 29, 29, 30, 30, 30, 31, 31, 32, 34, 34, 34, 34, 35, 35, 35, 36, 36

Offset: 0

N. J. A. Sloane, Nov 12 2006

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).

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

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.

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+2-i),i=1..nops(a)) ; end: A123577 := proc(n) L(n,5) ; end: for n from 0 to 80 do printf("%d, ",A123577(n)) ; od ; # R. J. Mathar, May 18 2007

(* The function L(n,t) is defined in A123575 *)
a[n_] := L[n, 5];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020 *)

More terms from R. J. Mathar, May 18 2007

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A123576 The Kruskal-Macaulay function L_4(n).

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