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A055248 Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)). 28
1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1, 512, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1024, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In the language of the Shapiro et al. reference (also given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-2*z)*(1-x*z/(1-z))).

a(n,m) = A008949(n,n-m), if n > m >= 0.

Binomial transform of the all 1's triangle: as a Riordan array, it factors to give (1/(1-x),x/(1-x))(1/(1-x),x). Viewed as a number square read by antidiagonals, it has T(n,k) = Sum_{j=0..n} binomial(n+k,n-j) and is then the binomial transform of the Whitney square A004070. - Paul Barry, Feb 03 2005

Riordan array (1/(1-2x), x/(1-x)). Antidiagonal sums are A027934(n+1), n >= 0. - Paul Barry, Jan 30 2005

Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674, ...); row sums of triangles A011971 and A159573. - Gary W. Adamson, Apr 16 2009

The matrix inverse starts

   1;

  -2,   1;

   2,  -3,   1;

  -2,   5,  -4,    1;

   2,  -7,   9,   -5,    1;

  -2,   9, -16,   14,   -6,    1;

   2, -11,  25,-  30,   20,   -7,    1;

  -2,  13, -36,   55,  -50,   27,   -8,    1;

   2, -15,  49,  -91,  105,  -77,   35,   -9,  1;

  -2,  17, -64,  140, -196,  182, -112,   44, -10,   1;

   2, -19,  81, -204,  336, -378,  294, -156,  54, -11, 1;

   ...

which may be related to A029653. - R. J. Mathar, Mar 29 2013

Read as a square array, this is the generalized Riordan array ( 1/(1 - 2*x), 1/(1 - x) ) as defined in the Bala link (p. 5), which factorizes as ( 1/(1 - x), x/(1 - x) )*( 1/(1 - x), x )*( 1, 1 + x ) = P*U*transpose(P), where P denotes Pascal's triangle, A007318, and U is the lower unit triangular array with 1's on or below the main diagonal. - Peter Bala, Jan 13 2016

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

P. Bala, Notes on generalized Riordan arrays

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

FORMULA

a(n, m) = Sum_{k=m..n} A007318(n, k) (partial row sums in columns m).

Column m recursion: a(n, m) = Sum_{j=m..n-1} a(j, m) + A007318(n, m) if n >= m >= 0, a(n, m) := 0 if n<m.

G.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.

a(n, m) = Sum_{j=0..n} binomial(n, m+j). - Paul Barry, Feb 03 2005

Inverse binomial transform (by columns) of A112626. - Ross La Haye, Dec 31 2006

T(2n,n) = A032443(n). - Philippe Deléham, Sep 16 2009

From Peter Bala, Dec 23 2014: (Start)

Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 7*x + 4*x^2/2! + x^3/3!) = 8 + 15*x + 26*x^2/2! + 42*x^3/3! + 64*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).

Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A143494 (but with a different offset). See the Example section. Cf. A106516. (End)

a(n,m) = Sum_{p=m..n} 2^(n-p)*binomial(p-1,m-1), n >= m >= 0, else 0. - Wolfdieter Lang, Jan 09 2015

T(n, k) = 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n+1], [n-k+2], 1/2). - Peter Luschny, Oct 10 2019

EXAMPLE

The triangle a(n,m) begins:

n\m    0    1    2   3   4   5   6   7  8  9 10 ...

0:     1

1:     2    1

2:     4    3    1

3:     8    7    4   1

4:    16   15   11   5   1

5:    32   31   26  16   6   1

6:    64   63   57  42  22   7   1

7:   128  127  120  99  64  29   8   1

8:   256  255  247 219 163  93  37   9  1

9:   512  511  502 466 382 256 130  46 10  1

10: 1024 1023 1013 968 848 638 386 176 56 11  1

... Reformatted. - Wolfdieter Lang, Jan 09 2015

Fourth row polynomial (n=3): p(3,x)= 8 + 7*x + 4*x^2 + x^3.

From Peter Bala, Dec 23 2014: (Start)

With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins

/1      \ /1        \ /1       \       /1       \

|2 1     ||0 1       ||0 1      |      |2  1     |

|4 3 1   ||0 2 1     ||0 0 1    |... = |4  5 1   |

|8 7 4 1 ||0 4 3 1   ||0 0 2 1  |      |8 19 9 1 |

|...     ||0 8 7 4 1 ||0 0 4 3 1|      |...      |

|...     ||...       ||...      |      |         |

= A143494. (End)

Matrix factorization of square array as P*U*transpose(P):

/1      \ /1        \ /1 1 1 1 ...\    /1  1  1  1 ...\

|1 1     ||1 1       ||0 1 2 3 ... |   |2  3  4  5 ... |

|1 2 1   ||1 1 1     ||0 0 1 3 ... | = |4  7 11 16 ... |

|1 3 3 1 ||1 1 1 1   ||0 0 0 1 ... |   |8 15 26 42 ... |

|...     ||...       ||...         |   |...            |

- Peter Bala, Jan 13 2016

MAPLE

T := (n, k) -> 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n + 1], [n-k + 2], 1/2).

seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Oct 10 2019

MATHEMATICA

a[n_, m_] := Sum[ Binomial[n, m + j], {j, 0, n}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Paul Barry *)

PROG

(Haskell)

a055248 n k = a055248_tabl !! n !! k

a055248_row n = a055248_tabl !! n

a055248_tabl = map reverse a008949_tabl

-- Reinhard Zumkeller, Jun 20 2015

CROSSREFS

Column sequences: A000079 (powers of 2, m=0), A000225 (m=1), A000295 (m=2), A002662 (m=3), A002663 (m=4), A002664 (m=5), A035038 (m=6), A035039 (m=7), A035040 (m=8), A035041 (m=9), A035042 (m=10).

Row sums: A001792(n) = A055249(n, 0).

Alternating row sums: A011782.

Cf. A011971, A159573. - Gary W. Adamson, Apr 16 2009

Cf. A007318, A008949, A106516, A143494.

Sequence in context: A134392 A048483 A276562 * A103316 A332389 A140069

Adjacent sequences:  A055245 A055246 A055247 * A055249 A055250 A055251

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang, May 26 2000

EXTENSIONS

Edited: Paul Barry, Jan 30 2005 comment changed. - Wolfdieter Lang, Jan 09 2015

STATUS

approved

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Last modified December 1 00:44 EST 2020. Contains 338831 sequences. (Running on oeis4.)