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A201509 From abs(A028297)=A034839*A007318 to A165241 via A113402. Second row (double triangle). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..48.

FORMULA

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

EXAMPLE

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

CROSSREFS

Cf. A039991, A034867.

Sequence in context: A049261 A135018 A073018 * A109295 A211188 A057899

Adjacent sequences:  A201506 A201507 A201508 * A201510 A201511 A201512

KEYWORD

nonn,tabf

AUTHOR

Paul Curtz, Dec 02 2011

STATUS

approved

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Last modified June 25 03:59 EDT 2022. Contains 354835 sequences. (Running on oeis4.)