A123572 -id:A123572 - cp's OEIS frontend

A123573 The Kruskal-Macaulay function K_4(n).

0, 4, 7, 9, 10, 10, 13, 15, 16, 16, 18, 19, 19, 20, 20, 20, 23, 25, 26, 26, 28, 29, 29, 30, 30, 30, 32, 33, 33, 34, 34, 34, 35, 35, 35, 35, 38, 40, 41, 41, 43, 44, 44, 45, 45, 45, 47, 48, 48, 49, 49, 49, 50, 50, 50, 50, 52, 53, 53, 54, 54, 54, 55, 55, 55, 55, 56, 56, 56, 56, 56

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 K_t(n) = C(n_t,t-1) + C(n_{t-1},t-2) + ... + 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 K_i(n), i=1, 2, 3, 4, 5 see A000012, A003057, A123572, A123573, A123574.

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

lowpol[n_, t_] := Module[{x}, x = Floor[(n*t!)^(1/t)]; While[Binomial[x, t] <= n, x = x + 1]; x - 1];
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];
K[n_, t_] := Module[{a}, a = c[n, t]; Sum[Binomial[a[[i]], t - i], {i, 1, Length[a]}]];
A123573[n_] := K[n, 4];
Table[A123573[n], {n, 0, 70}] (* Jean-François Alcover, Mar 30 2023, after R. J. Mathar *)

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

A123574 The Kruskal-Macaulay function K_5(n).

Original entry on oeis.org

0, 5, 9, 12, 14, 15, 15, 19, 22, 24, 25, 25, 28, 30, 31, 31, 33, 34, 34, 35, 35, 35, 39, 42, 44, 45, 45, 48, 50, 51, 51, 53, 54, 54, 55, 55, 55, 58, 60, 61, 61, 63, 64, 64, 65, 65, 65, 67, 68, 68, 69, 69, 69, 70, 70, 70, 70, 74, 77, 79, 80, 80, 83, 85, 86, 86, 88, 89, 89, 90

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 K_t(n) = C(n_t,t-1) + C(n_{t-1},t-2) + ... + 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 K_i(n), i=1, 2, 3, 4, 5 see A000012, A003057, A123572, A123573, A123574.

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: A123574 := proc(n) K(n,5) ; end: for n from 0 to 80 do printf("%d, ",A123574(n)) ; od ; # R. J. Mathar, May 18 2007

lowpol[n_, t_] := Module[{x}, x = Floor[(n*t!)^(1/t)]; While[Binomial[x, t] <= n, x = x + 1]; x - 1];
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];
K[n_, t_] := Module[{a}, a = c[n, t]; Sum[Binomial[a[[i]], t - i], {i, 1, Length[a]}]];
A123574[n_] := K[n, 5];
Table[A123574[n], {n, 0, 69}] (* Jean-François Alcover, Mar 30 2023, after R. J. Mathar *)

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

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A123573 The Kruskal-Macaulay function K_4(n).

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