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A125791 a(n) = 2^(n*(n-1)*(n-2)/6) for n>=1. 12
1, 1, 2, 16, 1024, 1048576, 34359738368, 72057594037927936, 19342813113834066795298816, 1329227995784915872903807060280344576, 46768052394588893382517914646921056628989841375232 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.

Also number of distinct instances of the one-in-three monotone 3SAT problem for n variables. - Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008

Hankel transform of aerated 2-Catalan numbers (A015083). [From Paul Barry, Dec 15 2010]

LINKS

Table of n, a(n) for n=0..10.

FORMULA

Determinant of n X n upper left corner submatrix of table A125790.

a(n) = 2^(binomial(1+n,n-2)). - Zerinvary Lajos, Jun 16 2007

EXAMPLE

a(n) is a pyramidal power of 2; exponents of 2 in a(n) begin:

[0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ..., n(n-1)(n-2)/6, ...].

MAPLE

seq(2^(binomial(1+n, n-2)), n=0..10); # Zerinvary Lajos, Jun 16 2007

PROG

(PARI) a(n)=if(n<1, 0, 2^(n*(n-1)*(n-2)/6))

(PARI) /* As determinant of n X n matrix: */

{a(n)=local(q=2, A=Mat(1), B); for(m=1, n, 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(matdet(matrix(n, n, r, c, (A^c)[r, 1])))}

for(n=1, 15, print1(a(n), ", "))

(Prolog program from Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008) This generates all 3SAT problem instances

test:-test(4).

test(Max):-

between(1, Max, N),

nl,

one_in_three_monotone_3sat(N, Pss),

write(N:Pss), nl,

fail

; nl.

% generates all one-in-three monotone 3SAT problems involving N variables

one_in_three_monotone_3sat(N, Pss):-

ints(1, N, Is),

findall(Xs, ksubset(3, Is, Xs), Xss),

subset_of(Xss, Pss).

% subset generator

subset_of([], []).

subset_of([X|Xs], Zs):-

subset_of(Xs, Ys),

add_element(X, Ys, Zs).

add_element(_, Ys, Ys).

add_element(X, Ys, [X|Ys]).

% subsets of K elements

ksubset(0, _, []).

ksubset(K, [X|Xs], [X|Rs]):-K>0, K1 is K-1, ksubset(K1, Xs, Rs).

ksubset(K, [_|Xs], Rs):-K>0, ksubset(K, Xs, Rs).

% list of integers in [From..To]

ints(From, To, Is):-findall(I, between(From, To, I), Is).

CROSSREFS

Cf. A125790, A078121; A002577, A000292 (pyramidal numbers).

Sequence in context: A084595 A273193 A002543 * A102103 A326974 A060597

Adjacent sequences:  A125788 A125789 A125790 * A125792 A125793 A125794

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 10 2006

EXTENSIONS

Name simplified; determinant formula moved out of name into formula section by Paul D. Hanna, Oct 16 2013

STATUS

approved

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Last modified January 24 01:05 EST 2020. Contains 331178 sequences. (Running on oeis4.)