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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)). 30
 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))). 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; edited by Wolfdieter Lang, Jan 09 2015 Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674, ...); row sums of triangles A011971 and A159573. - Gary W. Adamson, Apr 16 2009 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 Peter Bala, Notes on generalized Riordan arrays Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022). Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020. Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I. 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) = A008949(n, n-m), if n > m >= 0. 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= 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 T(n, k) = binomial(n, k)*hypergeom([1, k - n], [k + 1], -1). - Peter Luschny, Oct 06 2023 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. 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 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 *) T[n_, k_] := Binomial[n, k] * Hypergeometric2F1[1, k - n, k + 1, -1]; Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]] (* Peter Luschny, Oct 06 2023 *) 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 STATUS approved

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Last modified December 2 06:18 EST 2023. Contains 367509 sequences. (Running on oeis4.)