This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A161869 #9 Jul 08 2015 09:46:02 %S A161869 1,1,2,4,8,16,33,71,160,376,912,2256,5633,14093,35170,87344,215680, %T A161869 529568,1293633,3146515,7627208,18441476,44510160,107310480,258566402, %U A161869 622900466,1500717220,3616471960,8717948688,21023129472,50713990918,122374025914,295366777856 %N A161869 Convergent of an infinite product of Pascal's triangles aerated by rows. %C A161869 The sequence may be the binomial transform of A024493 interleaved with zeros. A024493 = (1, 1, 1, 2, 5, 11, 22,...); so the conjecture succeeds through a(12) = A007318 * [1, 0, 1, 0, 1, 0, 2, 0, 5, 0, 11,...]. %C A161869 Calculating more terms of the sequence shows the above conjecture is incorrect. - _Peter Bala_, Jul 07 2015 %F A161869 From _Peter Bala_, Jul 07 2015: (Start) %F A161869 Construct an infinite set of Pascal's triangles aerated by rows, denoted Pascal(1), Pascal(2), ..., where Pascal(1) = A007318, Pascal(2) is an aerated version of Pascal(1) with alternate rows 1, 3, 5, ... set equal to (0, 0, 0, ...), Pascal(3) is a further aeration of Pascal(1) with now two adjacent rows set equal to (0, 0, 0, ...), and so on. %F A161869 The infinite product Pascal(1)*Pascal(2)*Pascal(3)*... is well-defined. This sequence is the first column of the infinite product - all the other entries in the product are zero. (End) %e A161869 Pascal(1) Pascal(2) Pascal(3) %e A161869 1 1 1 %e A161869 1 1 0 0 0 0 %e A161869 1 2 1 1 1 0 0 0 0 %e A161869 1 3 3 1 0 0 0 0 1 1 0 0 %e A161869 1 4 6 4 1 1 2 1 0 0 0 0 0 0 0 %e A161869 ... %e A161869 First columns of %e A161869 Pascal(1): %e A161869 1, 1, 1, 1, 1, 1, 1, 1,... %e A161869 Pascal(1)*Pascal(2): %e A161869 1, 1, 2, 4, 8, 16, 32, 64,... %e A161869 Pascal(1)*Pascal(2)*Pascal(3): %e A161869 1, 1, 2, 4, 8, 16, 33, 71,... %e A161869 Pascal(1)*Pascal(2)*Pascal(3)*Pascal(4): %e A161869 1, 1, 2, 4, 8, 16, 33, 71,... %e A161869 ... %e A161869 converging to A161869. %p A161869 #A161869 %p A161869 #define aerated Pascal matrices (note indexing starts at 1) %p A161869 Pascal := proc (n) local i, j, r; %p A161869 Matrix(33, 33, (i, j) -> (product(r-(mod(i-1, n)), r = 1 .. n-1))*binomial(floor((i-1)/n), j-1) )/factorial(n-1) end proc: %p A161869 #it suffices to take the product of the first four aerated %p A161869 #Pascal arrays to get 33 correct terms of the sequence %p A161869 seq((Pascal(1).Pascal(2).Pascal(3).Pascal(4))(n, 1), n = 1 .. 33); %p A161869 # _Peter Bala_, Jul 07 2015 %Y A161869 Cf. A024493, A027826 (from first column of Pascal(2)^n as n -> inf). %K A161869 nonn,easy %O A161869 0,3 %A A161869 _Gary W. Adamson_, Jun 20 2009 %E A161869 Terms a(12) through a(32) added by _Peter Bala_, Jul 07 2015