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1, 1, 2, 2, 4, 5, 1, 8, 12, 4, 16, 28, 13, 1, 32, 64, 38, 6, 64, 144, 104, 25, 1, 128, 320, 272, 88, 8, 256, 704, 688, 280, 41, 1, 512, 1536, 1696, 832, 170, 10, 1024, 3328, 4096, 2352, 620, 61, 1, 2048, 7168
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OFFSET
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0,3
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COMMENTS
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Consider the irregular blocks of k verticals
1 1 2 1 4 3 8 8 1 16 20 5 32 48 18 1
1 1 2 2 4 5 1 8 12 4 16 28 13 1 32 64 38 6
2 3 1 4 7 3 8 16 9 1 16 36 25 5 32 80 66 19 1
4 9 6 1 8 20 16 4 16 44 41 14 1 32 96 102 44 6
8 24 25 10 1 16 52 61 30 5 32 112 146 85 20 1
16 60 85 55 15 1 32 128 198 146 50 6
32 144 258 231 105 21 1.
Without spaces the first row is abs(A028297) from Chebyshev polynomials. Note partial sums: 1,1,3,7,17,...=A001333.
First verticals are A113402. Vertical sums: 1,2,6,16,40,...=A057711(n+1)=A129952(n+1).
Algorithm: 1) The first 1 of row 1 (or first vertical) gives the second 1 of the second row.
2) 1 1 of the second vertical is the difference of the fifth vertical; 1 of third vertical is 1 of the sixth.
3) 2 2 2 of the fourth vertical gives 1 3 5 7 of the eighth; 1 2 3 of the fifth gives 0 (not written) 1 3 6 of the ninth; 1 of the sixth gives 1 of the tenth vertical.
A201509 is the pseudo-triangle whose successive lines are of the type T(n,0), T(n,1)+T(n-1,0), T(n,2)+T(n-1,1), ... T(n,k)+T(n-1,k-1), without 0's, with T=A201701. [e-mail, Philippe Deléham, Dec 04 2011]
a(n)=
1 1
2 2
4 5 1
8 12 4
16 28 13 1
32 64 38 6
64 144 104 25 1
128 320 272 88 8
Row sums=2,4,10,24=A052542(n+1).
Odd terms row sums: 1,2,5,12,29,70,...=A000129(n+1); also for even terms.
Terms of last row blocks: A165241.
Verticals: see A000079, A045623, A049611, A055585.
b(n)=
1,1,
2,2,0,
4,5,1,0,
8,12,4,0,0,
16,28,13,1,0,0,
could be considered.
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LINKS
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Table of n, a(n) for n=0..48.
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FORMULA
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T(n,k)= 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n<k. - Philippe Deléham, Dec 05 2011
The n-th row polynomial appears to equal sum {k = 1..floor((n+1)/2)} binomial(n,2*k-1)*(1+t)^k. Cf. A034867. - Peter Bala, Sep 10 2012
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EXAMPLE
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Triangle begins (full version):
0
1, 1
2, 2, 0
4, 5, 1, 0
8, 12, 4, 0, 0
16, 28, 13, 1, 0, 0
32, 64, 38, 6, 0, 0, 0
64, 144, 104, 25, 1, 0, 0, 0
128, 320, 272, 88, 8, 0, 0, 0, 0
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CROSSREFS
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Cf. A039991, A034867.
Sequence in context: A049261 A135018 A073018 * A109295 A211188 A057899
Adjacent sequences: A201506 A201507 A201508 * A201510 A201511 A201512
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KEYWORD
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nonn,tabf
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AUTHOR
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Paul Curtz, Dec 02 2011
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STATUS
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approved
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