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%I #80 Oct 06 2023 10:18:35
%S 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 128,127,120,99,64,29,8,1,256,255,247,219,163,93,37,9,1,512,511,502,
%U 466,382,256,130,46,10,1,1024,1023,1013,968,848,638,386,176,56,11,1
%N 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 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 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 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 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 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 Reinhard Zumkeller, <a href="/A055248/b055248.txt">Rows n = 0..125 of triangle, flattened</a>
%H Peter Bala, <a href="/A260492/a260492.pdf">Notes on generalized Riordan arrays</a>
%H 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 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 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 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 a(n, m) = A008949(n, n-m), if n > m >= 0.
%F a(n, m) = Sum_{k=m..n} A007318(n, k) (partial row sums in columns m).
%F 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 G.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.
%F a(n, m) = Sum_{j=0..n} binomial(n, m+j). - _Paul Barry_, Feb 03 2005
%F Inverse binomial transform (by columns) of A112626. - _Ross La Haye_, Dec 31 2006
%F T(2n,n) = A032443(n). - _Philippe Deléham_, Sep 16 2009
%F From _Peter Bala_, Dec 23 2014: (Start)
%F 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 Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
%F /I_k 0\
%F \ 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 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 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 T(n, k) = binomial(n, k)*hypergeom([1, k - n], [k + 1], -1). - _Peter Luschny_, Oct 06 2023
%e The triangle a(n,m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 2 1
%e 2: 4 3 1
%e 3: 8 7 4 1
%e 4: 16 15 11 5 1
%e 5: 32 31 26 16 6 1
%e 6: 64 63 57 42 22 7 1
%e 7: 128 127 120 99 64 29 8 1
%e 8: 256 255 247 219 163 93 37 9 1
%e 9: 512 511 502 466 382 256 130 46 10 1
%e 10: 1024 1023 1013 968 848 638 386 176 56 11 1
%e ... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
%e Fourth row polynomial (n=3): p(3,x)= 8 + 7*x + 4*x^2 + x^3.
%e The matrix inverse starts
%e 1;
%e -2, 1;
%e 2, -3, 1;
%e -2, 5, -4, 1;
%e 2, -7, 9, -5, 1;
%e -2, 9, -16, 14, -6, 1;
%e 2, -11, 25,- 30, 20, -7, 1;
%e -2, 13, -36, 55, -50, 27, -8, 1;
%e 2, -15, 49, -91, 105, -77, 35, -9, 1;
%e -2, 17, -64, 140, -196, 182, -112, 44, -10, 1;
%e 2, -19, 81, -204, 336, -378, 294, -156, 54, -11, 1;
%e ...
%e which may be related to A029653. - _R. J. Mathar_, Mar 29 2013
%e From _Peter Bala_, Dec 23 2014: (Start)
%e With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
%e /1 \ /1 \ /1 \ /1 \
%e |2 1 ||0 1 ||0 1 | |2 1 |
%e |4 3 1 ||0 2 1 ||0 0 1 |... = |4 5 1 |
%e |8 7 4 1 ||0 4 3 1 ||0 0 2 1 | |8 19 9 1 |
%e |... ||0 8 7 4 1 ||0 0 4 3 1| |... |
%e |... ||... ||... | | |
%e = A143494. (End)
%e Matrix factorization of square array as P*U*transpose(P):
%e /1 \ /1 \ /1 1 1 1 ...\ /1 1 1 1 ...\
%e |1 1 ||1 1 ||0 1 2 3 ... | |2 3 4 5 ... |
%e |1 2 1 ||1 1 1 ||0 0 1 3 ... | = |4 7 11 16 ... |
%e |1 3 3 1 ||1 1 1 1 ||0 0 0 1 ... | |8 15 26 42 ... |
%e |... ||... ||... | |... |
%e - _Peter Bala_, Jan 13 2016
%p T := (n,k) -> 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n + 1], [n-k + 2], 1/2).
%p seq(seq(simplify(T(n,k)), k=0..n),n=0..10); # _Peter Luschny_, Oct 10 2019
%t 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 T[n_, k_] := Binomial[n, k] * Hypergeometric2F1[1, k - n, k + 1, -1];
%t Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]] (* _Peter Luschny_, Oct 06 2023 *)
%o (Haskell)
%o a055248 n k = a055248_tabl !! n !! k
%o a055248_row n = a055248_tabl !! n
%o a055248_tabl = map reverse a008949_tabl
%o -- _Reinhard Zumkeller_, Jun 20 2015
%Y 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 Row sums: A001792(n) = A055249(n, 0).
%Y Alternating row sums: A011782.
%Y Cf. A011971, A159573. - _Gary W. Adamson_, Apr 16 2009
%Y Cf. A007318, A008949, A106516, A143494.
%K easy,nonn,tabl
%O 0,2
%A _Wolfdieter Lang_, May 26 2000
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