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%I #6 Nov 06 2020 09:23:07
%S 1,2,2,4,4,8,8,16,16,31,32,62,63,124,126,248,252,496,504,991,1007,
%T 1982,2013,3960,4023,7914,8040,15816,16068,31609,32112,63171,64180,
%U 126251,128266,252318,256347,504268,512324,1007801,1023909,2014131,2046338,4025329,4089724
%N Number of true-palindromic compositions of n.
%C A true-palindromic composition or true-palindrome to be a composition whose digit-comma-sequence is the same whether read from left to right or right to left. [Shapcott p. 35]
%H Caroline Shapcott, <a href="https://ajc.maths.uq.edu.au/pdf/60/ajc_v60_p035.pdf">An introduction to true-palindromic compositions</a>, Australasian Journal of Combinatorics, Volume 60(1) (2014), Pages 35-49.
%F Shapcott gives a g.f on p. 3, and 1 should be subtracted to get sequence for n>=1.
%e (12, 6, 21) is a true-palindromic composition of 39.
%e (126, 621) is a true-palindromic composition of 747.
%o (PARI) rev(n) = Vecrev(n=digits(n)); \\ A004086
%o ispal(n) = Vecrev(n=digits(n))==n; \\ A002113
%o radd(n) = fromdigits(Vecrev(digits(n))) + n; \\ A056964
%o lista(nn) = my(x='x+O('x^(nn))); Vec(sum(k=0, nn, if (ispal(k), x^k))/(1 - sum(k=1, nn, if (k%10, x^radd(k)))) - 1);
%Y Cf. A002113, A004086, A056964.
%Y Cf. A016116 (symmetric compositions), A338740.
%K nonn,base
%O 1,2
%A _Michel Marcus_, Nov 06 2020
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