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%I #10 May 01 2020 22:56:54
%S 1,1,2,14,222,10384,1633416,909329424,1817382187648,13096462193012288,
%T 341920990254898495104,32545025418981014632238080,
%U 11369364835492064157950365970944,14670198699267762936290991348116122624,70317372123398823244417008989595542434635776
%N The coefficient of x^(n + 2^n) in the n-th iteration of x^2+x^3 for n>=0.
%H Andrew Howroyd, <a href="/A251692/b251692.txt">Table of n, a(n) for n = 0..40</a>
%e The table of coefficients in iterations of x^2+x^3 begin:
%e [1];
%e [1, 1];
%e [1, 2, 2, 3, 3, 1];
%e [1, 4, 8, 14, 23, 32, 43, 63, 96, 141, 192, 240, 267, 255, 210, 150, 87, 36, 9, 1];
%e [1, 8, 32, 92, 222, 472, 906, 1626, 2810, 4746, 7886, 12908, ...];
%e [1, 16, 128, 696, 2940, 10384, 32036, 88804, 225752, 534788, ...];
%e [1, 32, 512, 5488, 44536, 293024, 1633416, 7953544, 34592576, ...];
%e [1, 64, 2048, 43744, 702448, 9056064, 97743376, 909329424, ...];
%e [1, 128, 8192, 349632, 11198432, 287200896, 6145427488, ...];
%e [1, 256, 32768, 2796416, 179011520, 9169571072, 391531929664, ...]; ...
%e in which this sequence forms the main diagonal.
%e The initial iterations of x^2+x^3 begin:
%e n=0: x;
%e n=1: x^2 + x^3;
%e n=2: x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + x^9;
%e n=3: x^8 + 4*x^9 + 8*x^10 + 14*x^11 + 23*x^12 + 32*x^13 + 43*x^14 + 63*x^15 + 96*x^16 + 141*x^17 + 192*x^18 + 240*x^19 + 267*x^20 + 255*x^21 + 210*x^22 + 150*x^23 + 87*x^24 + 36*x^25 + 9*x^26 + x^27;
%e n=4: x^16 + 8*x^17 + 32*x^18 + 92*x^19 + 222*x^20 + 472*x^21 + 906*x^22 + 1626*x^23 + 2810*x^24 + 4746*x^25 + 7886*x^26 + 12908*x^27 + 20752*x^28 +...;
%e ...
%o (PARI) a(n)={my(p=1 + O(x*x^n)); for(k=1, n, p = p^2 + if(2^(k-1)<=n, x^(2^(k-1))*p^3) ); polcoef(p, n)} \\ _Andrew Howroyd_, May 01 2020
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 02 2015
%E Terms a(11) and beyond from _Andrew Howroyd_, May 01 2020
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