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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 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.)