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%I #127 Mar 09 2024 08:14:58
%S 1,0,1,-1,0,1,0,-3,0,1,5,0,-6,0,1,0,25,0,-10,0,1,-61,0,75,0,-15,0,1,0,
%T -427,0,175,0,-21,0,1,1385,0,-1708,0,350,0,-28,0,1,0,12465,0,-5124,0,
%U 630,0,-36,0,1,-50521,0,62325,0,-12810,0,1050,0,-45,0,1
%N Exponential Riordan array (sech(x),x).
%C Row sums have e.g.f. exp(x)*sech(x) (signed version of A009006). Inverse of masked Pascal triangle A119467. Transforms the sequence with e.g.f. g(x) to the sequence with e.g.f. g(x)*sech(x).
%C Coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent and Bernoulli number (triangle read by rows). Another version in A153641. - _Philippe Deléham_, Oct 26 2013
%C Relations to Green functions and raising/creation and lowering/annihilation/destruction operators are presented in Hodges and Sukumar and in Copeland's discussion of this sequence and 2020 pdf. - _Tom Copeland_, Jul 24 2020
%H G. C. Greubel, <a href="/A119879/b119879.txt">Rows n=0..100 of triangle, flattened</a>
%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
%H T. Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera</a>, 2015.
%H T. Copeland, <a href="/A119879/a119879.txt">Discussion of this sequence</a>
%H T. Copeland, <a href="https://tcjpn.wordpress.com/2020/07/11/skipping-over-dimensions-juggling-zeros-in-the-matrix/">SKipping over Dimensions, Juggling Zeros in the Matrix</a>, 2020.
%H A. Hodges and C. V. Sukumar, <a href="http://dx.doi.org/10.1098/rspa.2007.0001">Bernoulli, Euler, permutations and quantum algebras</a>, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414.
%H Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifeDecompositions.html">Additive decompositions of classical numbers via the Swiss-Knife polynomials</a> [_Tom Copeland_, Oct 20 2015]
%H Miguel Méndez and Rafael Sánchez, <a href="https://arxiv.org/abs/1707.00336">On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops</a>, arXiv:1707.00336 [math.CO}, 2017, Section 4.3, Example 4.
%H Miguel A. Méndez and Rafael Sánchez Lamoneda, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i3p25">Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials</a>, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.
%F Number triangle whose k-th column has e.g.f. sech(x)*x^k/k!.
%F T(n,k) = C(n,k)*2^(n-k)*E_{n-k}(1/2) where C(n,k) is the binomial coefficient and E_{m}(x) are the Euler polynomials. - _Peter Luschny_, Jan 25 2009
%F The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n,k)*p{k}(0)*((n mod 2) - 1 + x^(n-k)). - _Peter Luschny_, Jul 16 2012
%F E.g.f.: exp(x*z)/cosh(x). - _Peter Luschny_, Aug 01 2012
%F Sum_{k=0..n} T(n,k)*x^k = A122045(n), A155585(n), A119880(n), A119881(n) for x = 0, 1, 2, 3 respectively. - _Philippe Deléham_, Oct 27 2013
%F With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of this entry, A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - _Tom Copeland_, Jan 24 2020
%F Triangle equals P*((I + P^2)/2)^(-1), where P denotes Pascal's triangle A007318. - _Peter Bala_, Mar 07 2024
%e Triangle begins:
%e 1;
%e 0, 1;
%e -1, 0, 1;
%e 0, -3, 0, 1;
%e 5, 0, -6, 0, 1;
%e 0, 25, 0, -10, 0, 1;
%e -61, 0, 75, 0, -15, 0, 1;
%e 0, -427, 0, 175, 0, -21, 0, 1;
%e 1385, 0, -1708, 0, 350, 0, -28, 0, 1;
%p T := (n,k) -> binomial(n,k)*2^(n-k)*euler(n-k,1/2): # _Peter Luschny_, Jan 25 2009
%t T[n_, k_] := Binomial[n, k] 2^(n-k) EulerE[n-k, 1/2];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2018, after _Peter Luschny_ *)
%o (Sage)
%o @CachedFunction
%o def A119879_poly(n, x) :
%o return 1 if n == 0 else add(A119879_poly(k, 0)*binomial(n, k)*(x^(n-k)-1+n%2) for k in range(n)[::2])
%o def A119879_row(n) :
%o R = PolynomialRing(ZZ, 'x')
%o return R(A119879_poly(n,x)).coeffs() # _Peter Luschny_, Jul 16 2012
%o # Alternatively:
%o (Sage) # uses[riordan_array from A256893]
%o riordan_array(sech(x), x, 9, exp=true) # _Peter Luschny_, Apr 19 2015
%o (PARI)
%o {T(n,k) = binomial(n,k)*2^(n-k)*(2/(n-k+1))*(subst(bernpol(n-k+1, x), x, 1/2) - 2^(n-k+1)*subst(bernpol(n-k+1, x), x, 1/4))};
%o for(n=0,5, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 25 2019
%Y Row sums are A155585. - _Johannes W. Meijer_, Apr 20 2011
%Y Rows reversed: A081658.
%Y Cf. A109449, A153641, A162660. - _Philippe Deléham_, Oct 26 2013
%Y Cf. A000182, A046802, A119467, A133314.
%Y Cf. A019538, A028246, A090582, A123125, A130850, A248727.
%K easy,sign,tabl
%O 0,8
%A _Paul Barry_, May 26 2006
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