Search: a166881 -id:a166881
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A166880
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Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of (x+x^2+x^3).
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+10
10
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1, 1, 1, 1, 1, 2, 4, 6, 8, 8, 6, 3, 1, 1, 3, 9, 24, 60, 138, 294, 579, 1053, 1767, 2739, 3924, 5196, 6352, 7152, 7389, 6969, 5961, 4587, 3144, 1896, 990, 438, 159, 45, 9, 1, 1, 4, 16, 60, 216, 744, 2460, 7818, 23910, 70446, 200160, 549006, 1455132, 3730846, 9262712
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OFFSET
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0,6
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LINKS
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EXAMPLE
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Triangle begins:
1;
1,1,1;
1,2,4,6,8,8,6,3,1;
1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1;
1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...;
1,9,81,720,6336,55224,477000,4085028,34700940,292495896,...;
1,10,100,990,9720,94680,915390,8787735,83795085,793894860,...;
1,11,121,1320,14300,153890,1645710,17494455,184915225,...;
1,12,144,1716,20328,239448,2805396,32700558,379309986,...;
1,13,169,2184,28080,359268,4575324,58009614,732380298,...;
1,14,196,2730,37856,522704,7188090,98465913,1343828395,...;
1,15,225,3360,49980,740670,10937010,160947465,2360704815,...;
1,16,256,4080,64800,1025760,16185840,254624520,3993857400,...;
1,17,289,4896,82688,1392368,23379216,391488648,6538326616,...;
1,18,324,5814,104040,1856808,33053814,586957419,10398271833,...;
...
The initial diagonals in this triangle begin:
A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
The diagonals are transformed one into the other by
1;
1,1;
3,2,1;
15,9,3,1;
114,62,18,4,1;
1159,593,157,30,5,1;
14838,7266,1812,316,45,6,1;
229401,108720,25989,4271,555,63,7,1;
4159662,1922166,445255,70180,8595,890,84,8,1; ...
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PROG
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(PARI) {T(n, k)=local(F=x+x^2+x^3, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A166884
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Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).
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+10
10
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1, 1, 1, 3, 2, 1, 15, 9, 3, 1, 114, 62, 18, 4, 1, 1159, 593, 157, 30, 5, 1, 14838, 7266, 1812, 316, 45, 6, 1, 229401, 108720, 25989, 4271, 555, 63, 7, 1, 4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1, 86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1
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OFFSET
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0,4
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LINKS
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EXAMPLE
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This triangle begins:
1;
1, 1;
3, 2, 1;
15, 9, 3, 1;
114, 62, 18, 4, 1;
1159, 593, 157, 30, 5, 1;
14838, 7266, 1812, 316, 45, 6, 1;
229401, 108720, 25989, 4271, 555, 63, 7, 1;
4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1;
86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1;
2034850425, 906623004, 201058614, 30000676, 3418245, 320070, 25963, 1912, 135, 10, 1;
53303009286, 23429034168, 5114874693, 748896765, 83336385, 7568355, 589057, 40882, 2631, 165, 11, 1; ...
Triangle A166880 of coefficients in iterations of x+x^2+x^3 begins:
1;
1,1,1;
1,2,4,6,8,8,6,3,1;
1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,...;
1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...; ...
in which this triangle transforms diagonals in A166880 into each other.
The initial diagonals in triangle A166880 begin:
A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
so that, if we treat the diagonals as column vectors, we have:
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PROG
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(PARI) {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A166882
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a(n) = coefficient of x^n in the n-th iteration of (x + x^2 + x^3) for n>=1.
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+10
4
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1, 2, 9, 60, 560, 6720, 98826, 1722084, 34700940, 793894860, 20329008975, 576026191026, 17893288364952, 604630781494558, 22079861395250568, 866509034147074284, 36367487433847501620, 1625458443704631873072
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OFFSET
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1,2
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LINKS
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EXAMPLE
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Let F_n(x) denote the n-th iteration of F(x) = x + x^2 + x^3;
then coefficients in the successive iterations of F(x) begin:
F(x):[(1), 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F_2: [1, (2), 4, 6, 8, 8, 6, 3, 1, 0, 0, ...];
F_3: [1, 3, (9), 24, 60, 138, 294, 579, 1053, 1767, 2739, ...];
F_4: [1, 4, 16, (60), 216, 744, 2460, 7818, 23910, 70446, 200160, ...];
F_5: [1, 5, 25, 120, (560), 2540, 11220, 48330, 203230, 835080, ...];
F_6: [1, 6, 36, 210, 1200, (6720), 36930, 199365, 1058175, ...];
F_7: [1, 7, 49, 336, 2268, 15078, (98826), 639093, 4080531, ...];
F_8: [1, 8, 64, 504, 3920, 30128, 228984, (1722084), 12821788, ...];
F_9: [1, 9, 81, 720, 6336, 55224, 477000, 4085028, (34700940), ...];
F_10:[1, 10, 100, 990, 9720, 94680, 915390, 8787735, 83795085, (793894860), ...]; ...
where the coefficients along the diagonal (shown above in parenthesis)
form the initial terms of this sequence.
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PROG
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(PARI) {a(n)=local(F=x+x^2+x^3, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n, x)))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A166883
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a(n) = coefficient of x^n in the (n+1)-th iteration of (x + x^2 + x^3) for n>=1.
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+10
4
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1, 3, 16, 120, 1200, 15078, 228984, 4085028, 83795085, 1943920935, 50333780640, 1439208976920, 45044270036220, 1531759925038616, 56239576979827360, 2217379518189430404, 93441321290076019236, 4191262657895865499821
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OFFSET
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1,2
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LINKS
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EXAMPLE
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Let F_n(x) denote the n-th iteration of F(x) = x + x^2 + x^3;
then coefficients in the successive iterations of F(x) begin:
F(x):[1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F_2: [(1), 2, 4, 6, 8, 8, 6, 3, 1, 0, 0, ...];
F_3: [1, (3), 9, 24, 60, 138, 294, 579, 1053, 1767, 2739, ...];
F_4: [1, 4, (16), 60, 216, 744, 2460, 7818, 23910, 70446, 200160, ...];
F_5: [1, 5, 25, (120), 560, 2540, 11220, 48330, 203230, 835080, ...];
F_6: [1, 6, 36, 210, (1200), 6720, 36930, 199365, 1058175, ...];
F_7: [1, 7, 49, 336, 2268, (15078), 98826, 639093, 4080531, ...];
F_8: [1, 8, 64, 504, 3920, 30128, (228984), 1722084, 12821788, ...];
F_9: [1, 9, 81, 720, 6336, 55224, 477000, (4085028), 34700940, ...];
F_10:[1, 10, 100, 990, 9720, 94680, 915390, 8787735, (83795085), ...]; ...
where the coefficients along the diagonal (shown above in parenthesis)
form the initial terms of this sequence.
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PROG
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(PARI) {a(n)=local(F=x+x^2+x^3, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n, x)))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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