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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)).

%I A055248 #88 May 06 2025 11:27:15
%S A055248 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,
%T A055248 128,127,120,99,64,29,8,1,256,255,247,219,163,93,37,9,1,512,511,502,
%U A055248 466,382,256,130,46,10,1,1024,1023,1013,968,848,638,386,176,56,11,1
%N 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)).
%C A055248 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))).
%C A055248 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
%C A055248 Riordan array (1/(1-2x), x/(1-x)). Antidiagonal sums are A027934(n+1), n >= 0. - _Paul Barry_, Jan 30 2005; edited by _Wolfdieter Lang_, Jan 09 2015
%C A055248 Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674, ...); row sums of triangles A011971 and A159573. - _Gary W. Adamson_, Apr 16 2009
%C A055248 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
%H A055248 Reinhard Zumkeller, <a href="/A055248/b055248.txt">Rows n = 0..125 of triangle, flattened</a>
%H A055248 Peter Bala, <a href="/A260492/a260492.pdf">Notes on generalized Riordan arrays</a>
%H A055248 Peter Bala, <a href="/A055248/a055248.pdf">A055248: Rapidly converging series for log(2) and Pi</a>
%H A055248 Jean-Luc Baril, Javier F. González, and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/BGR.pdf">Last symbol distribution in pattern avoiding Catalan words</a>, Univ. Bourgogne (France, 2022).
%H A055248 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H A055248 Norman Lindquist and Gerard Sierksma, <a href="https://doi.org/10.1016/0097-3165(81)90015-7">Extensions of set partitions</a>, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
%H A055248 L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.
%F A055248 a(n, m) = A008949(n, n-m), if n > m >= 0.
%F A055248 a(n, m) = Sum_{k=m..n} A007318(n, k) (partial row sums in columns m).
%F A055248 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.
%F A055248 G.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.
%F A055248 a(n, m) = Sum_{j=0..n} binomial(n, m+j). - _Paul Barry_, Feb 03 2005
%F A055248 Inverse binomial transform (by columns) of A112626. - _Ross La Haye_, Dec 31 2006
%F A055248 T(2n,n) = A032443(n). - _Philippe Deléham_, Sep 16 2009
%F A055248 From _Peter Bala_, Dec 23 2014: (Start)
%F A055248 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) ).
%F A055248 Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
%F A055248 /I_k 0\
%F A055248 \ 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)
%F A055248 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
%F A055248 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
%F A055248 T(n, k) = binomial(n, k)*hypergeom([1, k - n], [k + 1], -1). - _Peter Luschny_, Oct 06 2023
%F A055248 n-th row polynomial R(n, x) = (2^n - x*(1 + x)^n)/(1 - x). These polynomials can be used to find series acceleration formulas for the constants log(2) and Pi. - _Peter Bala_, Mar 03 2025
%e A055248 The triangle a(n,m) begins:
%e A055248 n\m    0    1    2   3   4   5   6   7  8  9 10 ...
%e A055248 0:     1
%e A055248 1:     2    1
%e A055248 2:     4    3    1
%e A055248 3:     8    7    4   1
%e A055248 4:    16   15   11   5   1
%e A055248 5:    32   31   26  16   6   1
%e A055248 6:    64   63   57  42  22   7   1
%e A055248 7:   128  127  120  99  64  29   8   1
%e A055248 8:   256  255  247 219 163  93  37   9  1
%e A055248 9:   512  511  502 466 382 256 130  46 10  1
%e A055248 10: 1024 1023 1013 968 848 638 386 176 56 11  1
%e A055248 ... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
%e A055248 Fourth row polynomial (n=3): p(3,x)= 8 + 7*x + 4*x^2 + x^3.
%e A055248 The matrix inverse starts
%e A055248    1;
%e A055248   -2,   1;
%e A055248    2,  -3,   1;
%e A055248   -2,   5,  -4,    1;
%e A055248    2,  -7,   9,   -5,    1;
%e A055248   -2,   9, -16,   14,   -6,    1;
%e A055248    2, -11,  25,-  30,   20,   -7,    1;
%e A055248   -2,  13, -36,   55,  -50,   27,   -8,    1;
%e A055248    2, -15,  49,  -91,  105,  -77,   35,   -9,  1;
%e A055248   -2,  17, -64,  140, -196,  182, -112,   44, -10,   1;
%e A055248    2, -19,  81, -204,  336, -378,  294, -156,  54, -11, 1;
%e A055248    ...
%e A055248 which may be related to A029653. - _R. J. Mathar_, Mar 29 2013
%e A055248 From _Peter Bala_, Dec 23 2014: (Start)
%e A055248 With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
%e A055248 /1      \ /1        \ /1       \       /1       \
%e A055248 |2 1     ||0 1       ||0 1      |      |2  1     |
%e A055248 |4 3 1   ||0 2 1     ||0 0 1    |... = |4  5 1   |
%e A055248 |8 7 4 1 ||0 4 3 1   ||0 0 2 1  |      |8 19 9 1 |
%e A055248 |...     ||0 8 7 4 1 ||0 0 4 3 1|      |...      |
%e A055248 |...     ||...       ||...      |      |         |
%e A055248 = A143494. (End)
%e A055248 Matrix factorization of square array as P*U*transpose(P):
%e A055248 /1      \ /1        \ /1 1 1 1 ...\    /1  1  1  1 ...\
%e A055248 |1 1     ||1 1       ||0 1 2 3 ... |   |2  3  4  5 ... |
%e A055248 |1 2 1   ||1 1 1     ||0 0 1 3 ... | = |4  7 11 16 ... |
%e A055248 |1 3 3 1 ||1 1 1 1   ||0 0 0 1 ... |   |8 15 26 42 ... |
%e A055248 |...     ||...       ||...         |   |...            |
%e A055248 - _Peter Bala_, Jan 13 2016
%p A055248 T := (n,k) -> 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n + 1], [n-k + 2], 1/2).
%p A055248 seq(seq(simplify(T(n,k)), k=0..n),n=0..10); # _Peter Luschny_, Oct 10 2019
%t A055248 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_ *)
%t A055248 T[n_, k_] := Binomial[n, k] * Hypergeometric2F1[1, k - n, k + 1, -1];
%t A055248 Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* _Peter Luschny_, Oct 06 2023 *)
%o A055248 (Haskell)
%o A055248 a055248 n k = a055248_tabl !! n !! k
%o A055248 a055248_row n = a055248_tabl !! n
%o A055248 a055248_tabl = map reverse a008949_tabl
%o A055248 -- _Reinhard Zumkeller_, Jun 20 2015
%Y A055248 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).
%Y A055248 Row sums: A001792(n) = A055249(n, 0).
%Y A055248 Alternating row sums: A011782.
%Y A055248 Cf. A011971, A159573. - _Gary W. Adamson_, Apr 16 2009
%Y A055248 Cf. A007318, A008949, A106516, A143494.
%K A055248 easy,nonn,tabl
%O A055248 0,2
%A A055248 _Wolfdieter Lang_, May 26 2000