A251692 - OEIS

%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