A275670 G.f. A(x,y) satisfies: A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1.
1, 0, 1, 0, 2, 0, 4, 0, 8, 1, 0, 16, 4, 0, 32, 14, 0, 64, 40, 0, 128, 108, 2, 0, 256, 272, 12, 0, 512, 664, 52, 0, 1024, 1568, 188, 0, 2048, 3632, 608, 1, 0, 4096, 8256, 1816, 12, 0, 8192, 18528, 5128, 76, 0, 16384, 41088, 13856, 360, 0, 32768, 90304, 36176, 1446, 0, 65536, 196864, 91856, 5192, 4, 0, 131072, 426368, 227968, 17192, 42, 0, 262144, 918016, 555040, 53504, 284, 0, 524288, 1966848, 1329696, 158588, 1496, 0, 1048576, 4195328, 3141632, 451824, 6704, 0, 2097152, 8914432, 7334208, 1245936, 26772, 6
Offset: 0
Examples
G.f.: A(x,y) = 1 + y*x + 2*y*x^2 + 4*y*x^3 + (y^2 + 8*y)*x^4 + (4*y^2 + 16*y)*x^5 + (14*y^2 + 32*y)*x^6 + (40*y^2 + 64*y)*x^7 + (2*y^3 + 108*y^2 + 128*y)*x^8 + (12*y^3 + 272*y^2 + 256*y)*x^9 + (52*y^3 + 664*y^2 + 512*y)*x^10 + (188*y^3 + 1568*y^2 + 1024*y)*x^11 + (y^4 + 608*y^3 + 3632*y^2 + 2048*y)*x^12 +...
such that A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1; further,
A(x,y) = x*y + ( x^2*y + A(x,x^2*y)^2 )^2,
A(x,y) = x*y + ( x^2*y + ( x^3*y + A(x,x^3*y)^2 )^2 )^2, etc.
This table of coefficients in g.f. A(x,y) begins:
1;
0, 1;
0, 2;
0, 4;
0, 8, 1;
0, 16, 4;
0, 32, 14;
0, 64, 40;
0, 128, 108, 2;
0, 256, 272, 12;
0, 512, 664, 52;
0, 1024, 1568, 188;
0, 2048, 3632, 608, 1;
0, 4096, 8256, 1816, 12;
0, 8192, 18528, 5128, 76;
0, 16384, 41088, 13856, 360;
0, 32768, 90304, 36176, 1446;
0, 65536, 196864, 91856, 5192, 4;
0, 131072, 426368, 227968, 17192, 42;
0, 262144, 918016, 555040, 53504, 284;
0, 524288, 1966848, 1329696, 158588, 1496;
0, 1048576, 4195328, 3141632, 451824, 6704;
0, 2097152, 8914432, 7334208, 1245936, 26772, 6;
0, 4194304, 18876416, 16943680, 3342784, 98060, 80;
0, 8388608, 39848960, 38785536, 8761720, 335704, 636;
0, 16777216, 83890176, 88063616, 22508448, 1088496, 3844;
0, 33554432, 176166912, 198506624, 56822624, 3375096, 19492;
0, 67108864, 369106944, 444562432, 141270272, 10080760, 87184, 4;
0, 134217728, 771764224, 989807872, 346507120, 29167000, 354628, 80;
0, 268435456, 1610629120, 2192154880, 839762496, 82113648, 1338376, 812;
0, 536870912, 3355467776, 4831741952, 2013427136, 225746384, 4753320, 5916;
0, 1073741824, 6979354624, 10603063808, 4781027584, 607828752, 16052296, 35000;
0, 2147483648, 14495563776, 23174734336, 11254280416, 1606760304, 51954808, 178904, 1; ...
Row polynomials begin:
n=0: 1;
n=1: y;
n=2: 2*y;
n=3: 4*y;
n=4: 8*y + y^2;
n=5: 16*y + 4*y^2;
n=6: 32*y + 14*y^2;
n=7: 64*y + 40*y^2;
n=8: 128*y + 108*y^2 + 2*y^3;
n=9: 256*y + 272*y^2 + 12*y^3;
n=10: 512*y + 664*y^2 + 52*y^3;
n=11: 1024*y + 1568*y^2 + 188*y^3;
n=12: 2048*y + 3632*y^2 + 608*y^3 + y^4;
n=13: 4096*y + 8256*y^2 + 1816*y^3 + 12*y^4;
n=14: 8192*y + 18528*y^2 + 5128*y^3 + 76*y^4;
n=15: 16384*y + 41088*y^2 + 13856*y^3 + 360*y^4;
n=16: 32768*y + 90304*y^2 + 36176*y^3 + 1446*y^4;
n=17: 65536*y + 196864*y^2 + 91856*y^3 + 5192*y^4 + 4*y^5; ...
the first row in which y^m appears is given by n = A033156(m), where A033156 begins:
[1, 4, 8, 12, 17, 22, 27, 32, 38, 44, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 200, ...].
Generating functions of initial columns.
G.f. of column 0: 1
G.f. of column 1: y*x/(1-2*x).
G.f. of column 2: y^2*x^4/((1-2*x)^2*(1-2*x^2)).
G.f. of column 3: y^3*2*x^8/((1-2*x)^3*(1-2*x^2)*(1-2*x^3)).
G.f. of column 4: y^4*x^12*(1 + 4*x - 10*x^3)/((1-2*x)^4*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)).
G.f. of column 5: y^5*x^17*(4 + 2*x + 8*x^2 - 28*x^4)/((1-2*x)^5*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)*(1-2*x^5)).
G.f. of column 6: y^6*x^22*(6 + 8*x - 20*x^3 - 24*x^4 - 36*x^5 - 56*x^6 + 16*x^7 + 176*x^8 + 224*x^9 - 336*x^11)/((1-2*x)^6*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)).
G.f. of column 7: y^7*x^27*(4 + 24*x + 4*x^2 - 12*x^3 - 72*x^5 - 112*x^6 - 96*x^7 + 112*x^8 - 64*x^9 + 64*x^10 + 496*x^11 + 576*x^12 - 1056*x^14) / ((1-2*x)^7*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)).
G.f. of column 8: y^8*x^32*(1 + 24*x + 36*x^2 - 4*x^3 - 88*x^4 - 202*x^5 - 14*x^6 - 82*x^7 - 168*x^8 + 400*x^9 + 440*x^10 + 892*x^11 + 1292*x^12 - 660*x^13 - 800*x^14 - 688*x^15 - 1776*x^16 - 1136*x^17 - 4504*x^18 - 2672*x^19 + 4672*x^20 + 5664*x^21 + 12672*x^22 - 13728*x^24) / ((1-2*x)^8*(1-2*x^2)^4*(1-2*x^3)^2*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)).
G.f. of column 9: y^9*x^38*(8 + 60*x + 72*x^2 + 16*x^3 - 238*x^4 - 584*x^5 - 232*x^6 + 172*x^7 + 328*x^8 + 52*x^9 + 1012*x^10 + 2636*x^11 + 1464*x^12 + 520*x^13 - 2040*x^14 - 664*x^15 - 2360*x^16 - 8712*x^17 - 13008*x^18 - 3696*x^19 + 12080*x^20 + 15392*x^21 + 1456*x^22 - 11040*x^23 + 18112*x^24 + 37728*x^25 + 47040*x^26 - 34304*x^27 - 78144*x^28 - 73216*x^29 + 91520*x^31) / ((1-2*x)^9*(1-2*x^2)^4*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)).
G.f. of column 10: y^10*x^44*(28 + 96*x + 198*x^2 - 160*x^3 - 864*x^4 - 596*x^5 - 856*x^6 - 384*x^7 + 3652*x^8 + 4752*x^9 + 696*x^10 - 2972*x^11 + 3928*x^12 + 4848*x^13 - 8360*x^14 - 18768*x^15 - 11000*x^16 - 14184*x^17 - 9896*x^18 + 17184*x^19 + 23664*x^20 + 7904*x^21 + 34480*x^22 + 53472*x^23 + 54160*x^24 + 68160*x^25 + 10560*x^26 - 166208*x^27 - 203488*x^28 - 86720*x^29 - 23552*x^30 + 13632*x^31 + 67584*x^32 - 95232*x^33 - 232256*x^34 + 129536*x^35 + 677632*x^36 + 624000*x^37 + 355840*x^38 - 67584*x^39 - 988416*x^40 - 1464320*x^41 + 1244672*x^43) / ((1-2*x)^10*(1-2*x^2)^5*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)^2*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)*(1-2*x^10)).
...
The g.f. of column n, y^n * x^A033156(n) * P(n,x)/Q(n,x), appears to have the following denominator:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k), with
P(n,x) being a polynomial of degree A024916(n) - A033156(n),
where A024916(n) = Sum_{k=1..n} k*floor(n/k).
...
Links
Programs
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PARI
/* Print first N rows of this triangle: */ N=32; {a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^2 + y*x^(n+1-k)); polcoeff(A, n)} {for(n=0, N, for(k=0,n, if(k==0,print1(polcoeff(a(n)+y*O(y^n),k,y)", "), if(polcoeff(a(n)+y*O(y^n),k,y)==0,break,print1(polcoeff(a(n)+y*O(y^n),k,y),", "))));print(""))}
Formula
G.f. A(x,y) satisfies: 1 = ...(((((A(x,y) - x*y)^(1/2) - x^2*y)^(1/2) - x^3*y)^(1/2) - x^4*y)^(1/2) - x^5*y)^(1/2) -...- x^n*y)^(1/2) -..., an infinite series of nested square roots.
Comments