%I #23 Sep 22 2024 04:42:53
%S 1,1,1,1,2,2,2,1,1,3,5,7,8,8,8,8,7,5,3,1,1,4,9,16,24,32,40,48,55,60,
%T 63,64,64,64,64,64,63,60,55,48,40,32,24,16,9,4,1,1,5,14,30,54,86,126,
%U 174,229,289,352,416,480,544,608,672,735,795,850,898,938,970,994,1010,1019
%N Triangle read by rows of [2^(n+1) - (n+1)] terms where row n has the g.f.: Product_{i=0..n-1} (1 + x^(2^i))^(n-i) for n>0, starting with a '1' in row 0.
%C Generating rule: Start with a single '1' in row 0; let S(n) denote the initial [2^(n+1) - (n+1)] terms of the partial sums of row n; generate row n+1 by concatenating the following: S(n), 2^[n*(n-1)/2] repeated (n-1) times and the terms of S(n) when read in reverse order.
%H Paul D. Hanna, <a href="/A131823/b131823.txt">Table of n, a(n) for n = 0..1990</a>
%H Jordan Stoyanov, Christophe Vignat, <a href="https://arxiv.org/abs/1901.04029">Non-Conventional Limits of Random Sequences Related to Partitions of Integers</a>, arXiv:1901.04029 [math.PR], 2019. [See page 2 and appendix for additional properties of this sequence.]
%H C. Vignat, T. Wakhare, <a href="https://arxiv.org/abs/1708.06479">Finite generating functions for the sum of digits sequence</a>, arXiv:1708.06479 [math.NT], 2017.
%H Tanay Wakhare, Christophe Vignat, <a href="https://arxiv.org/abs/1805.10569">Settling some sum suppositions</a>, arXiv:1805.10569 [math.NT], 2018.
%H Tanay Wakhare, Christophe Vignat, <a href="https://doi.org/10.1007/s10474-018-0886-8">Settling some sum suppositions</a>, Acta Math. Hungar. (2018).
%F Row sums are 2^(n*(n+1)/2) for n>=0.
%e Triangle begins:
%e 1;
%e 1,1;
%e 1,2, 2, 2,1;
%e 1,3,5,7,8, 8,8, 8,7,5,3,1;
%e 1,4,9,16,24,32,40,48,55,60,63,64, 64,64,64, 64,63,60,55,48,40,32,24,16,9,4,1; ...
%e Illustrate the row g.f.s by:
%e (1+x)^2*(1+x^2) = g.f. of row 2: [1,2,2,2,1];
%e (1+x)^3*(1+x^2)^2*(1+x^4) = g.f. of row 3: [1,3,5,7,8,8,8,8,7,5,3,1];
%e (1+x)^4*(1+x^2)^3*(1+x^4)^2*(1+x^8) = g.f. of row 4.
%t Rest@ Flatten@ Array[{1}~Join~CoefficientList[Series[Product[(1 + x^(2^i))^(# - i), {i, 0, # - 1}], {x, 0, 2^(# + 1) - (# + 1)}], x] &, 5] (* _Michael De Vlieger_, Aug 21 2018 *)
%o (PARI) {T(n,k)=local(A=[1]);if(n==0,1,for(i=0,n-1, A=concat(Vec((Polrev(A)+O(x^(#A+i)))/(1-x)),Vec(O(x^(#A))+Pol(Vec(Ser(A)/(1-x))))));A[k+1])}
%o for(n=0,6,for(k=0,2^(n+1)-(n+2), print1(T(n,k),", "));print(""))
%Y Cf. A131824 (main diagonal).
%K nonn,tabf
%O 0,5
%A _Paul D. Hanna_, Jul 19 2007