A262975 Union of all unique coefficients of all powers of the g.f. A(x) of this sequence, starting with A(0)=2 and A'(0)=3.

2, 3, 4, 8, 12, 16, 25, 32, 36, 56, 64, 96, 102, 112, 128, 200, 240, 256, 267, 344, 356, 384, 512, 576, 598, 636, 656, 936, 1024, 1040, 1048, 1128, 1344, 1368, 1480, 1952, 2048, 2248, 2800, 2865, 2884, 2928, 3072, 3200, 3360, 3640, 4096, 4905, 5408, 5460, 5760, 6912, 6948, 7088, 7840, 8192, 8736, 9688, 10083, 11088, 11210, 11616, 13552, 14208, 15360, 16384, 16608, 18056, 18576, 20224, 21020, 22800, 24992, 31123, 31356, 32768, 33368, 33792, 34832, 35872

Offset

0,1

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Example

G.f.: A(x) = 2 + 3*x + 4*x^2 + 8*x^3 + 12*x^4 + 16*x^5 + 25*x^6 + 32*x^7 + 36*x^8 + 56*x^9 + 64*x^10 + 96*x^11 + 102*x^12 +...
The coefficients in A(x)^n begin:
[2, 3, 4, 8, 12, 16, 25, 32, 36, 56, 64, 96, 102, 112, 128, 200, ...];
[4, 12, 25, 56, 112, 200, 356, 598, 936, 1480, 2248, 3360, 4905, ...];
[8, 36, 102, 267, 636, 1368, 2800, 5460, 10083, 18056, 31356, ...];
[16, 96, 344, 1048, 2865, 7088, 16384, 35872, 74584, 148876,  ...];
[32, 240, 1040, 3640, 11210, 31123, 79940, 193160, 442420,  ...];
[64, 576, 2928, 11616, 39804, 122148, 344329, 907656, 2260656,  ...];
[128, 1344, 7840, 34832, 131544, 441532, 1353198, 3858011,  ...];
[256, 3072, 20224, 99584, 411232, 1497920, 4954608, 15175216,  ...];
[512, 6912, 50688, 274176, 1229760, 4830624, 17142816, 56099376,  ...];
[1024, 15360, 124160, 732160, 3546240, 14943872, 56621600,  ...]; ...
where the sorted union of all unique coefficients forms this sequence.
...
The coefficients of A(x)^n are located at the following positions:
n=2: [3,5,7,10,14,16,21,25,28,35,38,45,48,53,58,63,69,73,77,85,89,94,102,,...].
n=4: [4,9,13,19,26,34,39,50,59,68,75,87,100,112,124,135,149,162,176,189,204,...].
n=5: [6,12,20,31,40,54,66,80,95,114,130,152,170,190,209,236,254,275,298,322,...].
n=6: [8,17,30,46,61,74,98,120,145,169,196,223,251,285,314,341,377,409,446,476,...].
n=7: [11,24,42,62,84,106,138,166,200,238,269,308,349,384,430,470,520,570,625,...].
n=8: [18,43,70,104,143,184,234,282,332,385,448,509,576,653,732,808,890,974,...].
n=9: [23,52,86,129,177,230,288,345,411,475,554,637,726,815,909,1005,...].
n=10: [29,65,108,159,217,280,347,422,495,584,680,777,878,990,...].
...
The powers of 2 are located at positions:
[1,3,4,6,8,11,15,18,23,29,37,47,56,66,76,91,110,128,150,172,197,224,253,287,321,359,395,437,482,535,591,652,718,783,863,936,1015,...].

Prog

(PARI) {a(n)=local(A=[2, 3], G=A); for(i=1, sqrt(n+1), G=A; for(k=2, 4*#binary(n), G=concat(G, Vec(Ser(A)^k)); G=vecsort(G, , 8) ); A=Vec(Ser(G) +x*O(x^n)) ); A[n]}
for(n=1, 80, print1(a(n), ", "))

Crossrefs

Sequence in context: A030073 A115271 A189375 * A062923 A360363 A133464

Adjacent sequences: A262972 A262973 A262974 * A262976 A262977 A262978

Keyword

nonn

Author

Paul D. Hanna, Oct 05 2015

Status

approved