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%I #88 May 02 2023 14:41:56
%S 1,1,1,3,4,1,15,23,9,1,105,176,86,16,1,945,1689,950,230,25,1,10395,
%T 19524,12139,3480,505,36,1,135135,264207,177331,57379,10045,973,49,1,
%U 2027025,4098240,2924172,1038016,208054,24640,1708,64,1,34459425,71697105,53809164,20570444,4574934,626934,53676,2796,81,1
%N Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
%C Exponential Riordan array (1/sqrt(1-2*x), log(1/sqrt(1-2*x))). - _Paul Barry_, May 09 2011
%C The o.g.f.s D(d, x) of the column sequences, for d, d >= 0,(d=0 for the main diagonal) are P(d, x)/(1 - x)^(2*d+1), with the row polynomial P(d, x) = Sum_{m=0..d} A288875(d, m)*x^m. See A288875 for details. - _Wolfdieter Lang_, Jul 21 2017
%H T. D. Noe, <a href="/A028338/b028338.txt">Rows n=0..50 of triangle, flattened</a>
%H Priyavrat Deshpande, Krishna Menon, and Anurag Singh, <a href="https://arxiv.org/abs/2103.03865">A combinatorial statistic for labeled threshold graphs</a>, arXiv:2103.03865 [math.CO], 2021.
%H Thomas Godland and Zakhar Kabluchko, <a href="https://arxiv.org/abs/2009.04186">Projections and angle sums of permutohedra and other polytopes</a>, arXiv:2009.04186 [math.MG], 2020.
%H Thomas Godland and Zakhar Kabluchko, <a href="https://doi.org/10.1007/s00025-023-01918-2">Projections and Angle Sums of Belt Polytopes and Permutohedra</a>, Res. Math. (2023) Vol. 78, Art. No. 140.
%H Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, <a href="http://arxiv.org/abs/1510.04073">Convex hulls of random walks, hyperplane arrangements, and Weyl chambers</a>, arXiv preprint arXiv:1510.04073 [math.PR], 2015.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H Bruce E. Sagan and Joshua P. Swanson, <a href="https://arxiv.org/abs/2205.14078">q-Stirling numbers in type B</a>, arXiv:2205.14078 [math.CO], 2022.
%F Triangle T(n, k), read by rows, given by [1, 2, 3, 4, 5, 6, 7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 20 2005
%F T(n, k) = Sum_{i=k..n} (-2)^(n-i) * binomial(i, k) * s(n, i) where s(n, k) are signed Stirling numbers of the first kind. - Francis Woodhouse (fwoodhouse(AT)gmail.com), Nov 18 2005
%F G.f. of row polynomials in y: 1/(1-(x+x*y)/(1-2*x/(1-(3*x+x*y)/(1-4*x/(1-(5*x+x*y)/(1-6*x*y/(1-... (continued fraction). - _Paul Barry_, Feb 07 2009
%F T(n, m) = (2*n-1)*T(n-1,m) + T(n-1,m-1) with T(n, 0) = (2*n-1)!! and T(n, n) = 1. - _Johannes W. Meijer_, Jun 08 2009
%F From _Wolfdieter Lang_, May 09 2017: (Start)
%F E.g.f. of row polynomials in y: (1/sqrt(1-2*x))*exp(-y*log(sqrt(1-2*x))) = exp(-(1+y)*log(sqrt(1-2*x))) = 1/sqrt(1-2*x)^(1+y).
%F E.g.f. of column m sequence: (1/sqrt(1-2*x))* (-log(sqrt(1-2*x)))^m/m!. For the special Sheffer, also known as exponential Riordan array, see a comment above. (End)
%F Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 2^(n-1-p)*(1 + 2*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1). See a comment and references in A286718. - _Wolfdieter Lang_, Aug 09 2017
%e G.f. for n = 4: (x + 1)*(x + 3)*(x + 5)*(x + 7) = 105 + 176*x + 86*x^2 + 16*x^3 + x^4.
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9
%e 0: 1
%e 1: 1 1
%e 2: 3 4 1
%e 3: 15 23 9 1
%e 4: 105 176 86 16 1
%e 5: 945 1689 950 230 25 1
%e 6: 10395 19524 12139 3480 505 36 1
%e 7: 135135 264207 177331 57379 10045 973 49 1
%e 8: 2027025 4098240 2924172 1038016 208054 24640 1708 64 1
%e 9: 34459425 71697105 53809164 20570444 4574934 626934 53676 2796 81 1
%e ...
%e row n = 10: 654729075 1396704420 1094071221 444647600 107494190 16486680 1646778 106800 4335 100 1.
%e ... reformatted and extended. - _Wolfdieter Lang_, May 09 2017
%e O.g.f.s of diagonals d >= 0: D(2, x) = (3 + 8*x + x^2)/(1 - x)^5 generating [3, 23, 86, ...] = A024196(n+1), from the row d=2 entries of A288875 [3, 8, 1]. - _Wolfdieter Lang_, Jul 21 2017
%e Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(2*(1+4*(5/12 )*T(2, 2)/2! + 1*(1 + 4*(1/2))*T(3,2)/3!) = (4!/2)*(8/3*1 + 3*9/3!) = 86. - _Wolfdieter Lang_, Aug 11 2017
%p nmax:=8; for n from 0 to nmax do a(n, 0) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (2*n-1)*a(n-1, m) + a(n-1, m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # _Johannes W. Meijer_, Jun 08 2009, revised Nov 25 2012
%t T[n_, k_] := Sum[(-2)^(n-i) Binomial[i, k] StirlingS1[n, i], {i, k, n}] (* Woodhouse *)
%t Join[{1},Flatten[Table[CoefficientList[Expand[Times@@Table[x+i,{i,1,2n+1,2}]],x],{n,0,10}]]] (* _Harvey P. Dale_, Jan 29 2013 *)
%Y A039757 is signed version.
%Y Row sums: A000165.
%Y Columns: A001147, A004041, A028339, A028340, A028341; A000012, A000290, A024196, A024197, A024198.
%Y Diagonals: A000012, A000290(n+1), A024196(n+1), A024197(n+1), A024198(n+1).
%Y A161198 is a scaled triangle version and A109692 is a transposed triangle version.
%Y Central terms: A293318.
%Y Cf. A286718, A002208(n+1)/A002209(n+1).
%K tabl,nonn,easy,nice
%O 0,4
%A _Bill Gosper_
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