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A125792
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Column 2 of table A125790; also equals row sums of matrix power A078121^2.
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12
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1, 3, 9, 35, 201, 1827, 27337, 692003, 30251721, 2320518947, 316359580361, 77477180493603, 34394869942983369, 27893897106768940835, 41603705003444309596873, 114788185359199234852802339, 588880400923055731115178072777, 5642645813427132737155703265972003
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OFFSET
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0,2
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COMMENTS
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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]
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LINKS
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FORMULA
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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]
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)
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EXAMPLE
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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]
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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