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A123578 The Kruskal-Macaulay function M_2(n). 7
0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Identical to A002024, except for the initial 0.

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).

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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:

M := proc(n, t) local a ; a := C(n, t) ; add( binomial(op(i, a)-1, t-i), i=1..nops(a)) ; end:

A123578 := proc(n) M(n, 2) ; end: # R. J. Mathar, Mar 14 2007

MATHEMATICA

lowpol[n_, t_] := Module[{x = Floor[(n*t!)^(1/t)]}, While[Binomial[x, t] <= n, x = x+1]; x-1]; c[n_, t_] := Module[{nresid = n, tresid = t, a = {}, m}, While[nresid > 0, m = lowpol[nresid, tresid]; AppendTo[a, m]; nresid = nresid - Binomial[m, tresid]; tresid = tresid-1]; a]; m[n_, t_] := With[{a = c[n, t]}, Sum[ Binomial[ a[[i]]-1, t-i], {i, 1, Length[a]}]]; a[n_] := m[n, 2]; Table[a[n], {n, 0, 84}] (* Jean-Fran├žois Alcover, Dec 04 2012, translated from R. J. Mathar's Maple program *)

PROG

(PARI) A123578(n)=(sqrtint(8*n)+1)\2 \\ M. F. Hasler, Apr 19 2014

CROSSREFS

For M_i(n), i=1, 2, 3, 4, 5 see A000127, A123578, A123579, A123580, A123731.

Sequence in context: A087847 A107436 A002024 * A087845 A130146 A113764

Adjacent sequences:  A123575 A123576 A123577 * A123579 A123580 A123581

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 12 2006

EXTENSIONS

More terms from R. J. Mathar, Mar 14 2007

STATUS

approved

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Last modified December 4 05:11 EST 2016. Contains 278748 sequences.