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A105851 Binomial transform triangle, read by rows. 2
1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 12, 5, 1, 32, 48, 32, 16, 6, 1, 64, 112, 80, 44, 20, 7, 1, 128, 256, 192, 112, 56, 24, 8, 1, 256, 576, 448, 272, 144, 68, 28, 9, 1, 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1, 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Let P = Pascal's triangle as an infinite lower triangular matrix and A = the infinite array of arithmetic sequences as shown in A077028:

1 1 1 1 1...

1 2 3 4 5...

1 3 5 7 9...

1 4 7 10 13...

1 5 9 13 17...

Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2...).

From Peter Bala, Jul 26 2015: (Start)

T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.

O.g.f.: (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....

k-th column g.f.: (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)

EXAMPLE

Column 3: 1, 5, 16, 44, 112...(A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7,...).

Triangle begins:

1;

2, 1;

4, 3, 1;

8, 8, 4, 1;

16, 20, 12, 5, 1;

32, 48, 32, 16, 6, 1;

64, 112, 80, 44, 20, 7, 1;

128, 256, 192, 112, 56, 24, 8, 1;

256, 576, 448, 272, 144, 68, 28, 9, 1;

512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1;

1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1 ;...

MAPLE

seq(seq((2 + k*(n - k))*2^(n-k-1), k=0..n), n=0..10); # Peter Bala, Jul 26 2015

MATHEMATICA

t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jul 26 2015 *)

PROG

(MAGMA) /* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 26 2015

CROSSREFS

Cf. A077028, A001792, A001787, A053220, A016777, A014480.

Sequence in context: A055248 A103316 A140069 * A106195 A247023 A051129

Adjacent sequences:  A105848 A105849 A105850 * A105852 A105853 A105854

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Apr 23 2005

EXTENSIONS

More terms from Philippe Deléham, Mar 31 2007

STATUS

approved

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Last modified March 25 03:50 EDT 2019. Contains 321450 sequences. (Running on oeis4.)