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A233968 Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.

%I A233968 #8 Jan 18 2014 16:21:49
%S A233968 2,4,6,12,16,30,38,64,84,128,166,248,314,448,576,790,1004,1358,1708,
%T A233968 2264,2844,3694,4614,5936,7354,9342,11544,14502,17816,22220,27144,
%U A233968 33584,40878,50192,60828,74276,89596,108778,130772,157918,189116,227374
%N A233968 Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.
%C A233968 Also first differences of A211978.
%F A233968 a(n) = 2*(A006128(n) - A006128(n-1)) = 2*A138137(n).
%e A233968 Illustration of initial terms as a dissection of a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
%e A233968 .                                         _ _ _ _ _ _
%e A233968 .                                         _ _ _      |
%e A233968 .                                         _ _ _|_    |
%e A233968 .                                         _ _    |   |
%e A233968 .                             _ _ _ _ _      |   |   |
%e A233968 .                             _ _ _    |             |
%e A233968 .                   _ _ _ _        |   |             |
%e A233968 .                   _ _    |           |             |
%e A233968 .           _ _ _      |   |           |             |
%e A233968 .     _ _        |         |           |             |
%e A233968 . _      |       |         |           |             |
%e A233968 .  |     |       |         |           |             |
%e A233968 .
%e A233968 . 2    4      6       12          16          30
%e A233968 .
%e A233968 Also using the elements from the above diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
%e A233968 7..................................
%e A233968 .                                 /\
%e A233968 5....................            /  \                /\
%e A233968 .                   /\          /    \          /\  /
%e A233968 3..........        /  \        /      \        /  \/
%e A233968 2.....    /\      /    \    /\/        \      /
%e A233968 1..  /\  /  \  /\/      \  /            \  /\/
%e A233968 0 /\/  \/    \/          \/              \/
%e A233968 .  2, 4,   6,       12,           16,...
%e A233968 .
%Y A233968 Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228109, A228110, A229946.
%K A233968 nonn
%O A233968 1,1
%A A233968 _Omar E. Pol_, Jan 14 2014