OFFSET
0,2
COMMENTS
Triangle A078121 shifts left one column under matrix square and is related to partitions into powers of 2.
Number of partitions of 2^n into powers of 2, excluding the trivial partition 2^n=2^n. - Valentin Bakoev, Feb 15 2009
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..86
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.
FORMULA
Is this sequence the same as A002575 (coefficients of Bell's formula)?
Denote the sum m^n + m^n + ... + m^n, k times, by k*m^n (m > 1, n > 0 and k are natural numbers). The general formula for the number of all partitions of the sum k*m^n into powers of m, smaller than m^n, is t_m(n, k)= 1 when n=1 or k=0, or = t_m(n, k-1) + Sum_{j=1..m} t_m(n-1, (k-1)*n+j), when n > 1 and k > 0. A125792 is obtained for m=2 and n=1,2,3,... [Valentin Bakoev, Feb 15 2009]
a(n) = A145515(n+1,2)-1. - Alois P. Heinz, Feb 27 2009
From Benedict W. J. Irwin, Nov 16 2016: (Start)
Conjecture: a(n+1) = Sum_{i_1=1..3} Sum_{i_2=1..2*i_1-1} ... Sum_{i_n=1..2*i_(n-1)-1} (2*i_n - 1). For example:
a(2) = Sum_{i=1..3} 2*i-1.
a(3) = Sum_{i=1..3} Sum_{j=1..2*i-1} 2*j-1.
a(4) = Sum_{i=1..3} Sum_{j=1..2*i-1} Sum_{k=1..2*j-1} 2*k-1. (End)
EXAMPLE
G.f.: 1 + 3*x + 9*x^2 + 35*x^3 + 201*x^4 + 1827*x^5 + 27337*x^6 + 692003*x^7 + ...
To obtain t_2(5,1) we use the table T, defined as T[i,j]= t_2(i,j), for i=1,2,...,5(=n), and j= 0,1,2,...,16(= k*m^{n-1}). It is 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 1,3,5,7,9,11,13,15,17 1,9,25,49,81 1,35,165 1,201 Column 1 contains the first 5 members of A125792. [Valentin Bakoev, Feb 15 2009]
MAPLE
g:= proc(b, n, k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1, n+1, 2)-1: seq(a(n), n=0..25); # Alois P. Heinz, Feb 27 2009
MATHEMATICA
T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, _] = T[_, 0] = 1; Table[T[n, 2], {n, 0, 20} ] (* Jean-François Alcover, Jun 15 2015 *)
PROG
(PARI) {a(n)=local(p=2, q=2, A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i||j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0, n, (A^p)[n+1, c+1]))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n, k=3) = if(n<1, n==0, sum(i=1, k, a(n-1, 2*i-1)))}; /* Michael Somos, Nov 24 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 10 2006
STATUS
approved