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%I #34 Mar 10 2024 03:41:01
%S 0,1,0,0,2,0,-2,0,3,0,0,-8,0,4,0,16,0,-20,0,5,0,0,96,0,-40,0,6,0,-272,
%T 0,336,0,-70,0,7,0,0,-2176,0,896,0,-112,0,8,0,7936,0,-9792,0,2016,0,
%U -168,0,9,0,0,79360,0,-32640,0,4032,0,-240,0,10,0
%N Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.
%C Definition. V_n(x) = (skp(n, x+1) - skp(n, x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - _Peter Luschny_, Jul 23 2012
%C Equivalently, let the polynomials V_n(x) (n>=0) defined by V_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*L(k)*(x+v+1)^n; the sequence L(k) = -1 - H(k-1)*(-1)^floor((k-1)/4) / 2^floor(k/2) if k > 0 and L(0)=0; H(k) = 1 if k mod 4 <> 0, otherwise 0.
%C (1) V_n(0) = 2^n * Euler(n,1) for n > 0, A155585.
%C (2) V_n(1) = 1 - Euler(n).
%C (3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642.
%C (4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1.
%C (5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586.
%C The Swiss-Knife polynomials (A153641) are complementary to the polynomials defined here. Adding both gives polynomials with e.g.f. exp(x*t)*(sech(t)+tanh(t)), the coefficients of which are a signed variant of A109449.
%C The Swiss-Knife polynomials as well as the complementary Swiss-Knife polynomials are closely related to the Bernoulli and Euler polynomials. Let F be a sequence and
%C P_{F}[n](x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n.
%C V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials,
%C W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials.
%C B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials,
%C E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials.
%C The most striking formal difference between the Swiss-Knife-type polynomials and the Bernoulli-Euler type polynomials is: The SK-type polynomials have integer coefficients whereas the BE-type polynomials have rational coefficients.
%C Let R be the exponential Riordan array (exp(x)*sech(x), x) = P * A119879 = 2*P(I + P^2)^(-1) where P denotes Pascal's triangle A007318. Then T = R - I. - _Peter Bala_, Mar 07 2024
%H Leonhard Euler (1735), <a href="http://arxiv.org/abs/math/0506415">De summis serierum reciprocarum</a>, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
%H Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials</a>.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SwissKnifePolynomials">Swiss-Knife Polynomials and Euler Numbers</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>.
%H J. Worpitzky, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002158698">Studien über die Bernoullischen und Eulerschen Zahlen</a>, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.
%F T(n, k) = [x^(n-k)](skp(n,x+1)-skp(n,x-1))/2) where skp(n,x) are the Swiss-Knife polynomials A153641. - _Peter Luschny_, Jul 23 2012
%F E.g.f. exp(x*t)*tanh(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2-2)*(t^3/3!) + ...
%F V_n(x) = -x^n + Sum_{k=0..n} C(n,k)*Euler(k)*(x+1)^(n-k).
%e Triangle begins:
%e [0] 0;
%e [1] 1, 0;
%e [2] 0, 2, 0;
%e [3] -2, 0, 3, 0;
%e [4] 0, -8, 0, 4, 0;
%e [5] 16, 0, -20, 0, 5, 0;
%e [6] 0, 96, 0, -40, 0, 6, 0;
%e [7] -272, 0, 336, 0, -70, 0, 7, 0;
%e [8] 0, -2176, 0, 896, 0, -112, 0, 8, 0;
%e [9] 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0;
%p # Polynomials V_n(x):
%p V := proc(n,x) local k,pow; pow := (n,k) -> `if`(n=0 and k=0,1,n^k); add(binomial(n,k)*euler(k)*pow(x+1,n-k),k=0..n) - pow(x,n) end:
%p # Coefficients a(n):
%p seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t),t,16),t,n),x,k),k=0..n)),n=0..8);
%t skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; v[n_, x_] := (skp[n, x+1]-skp[n, x-1])/2; t[n_, k_] := Coefficient[v[n, x], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2014 *)
%o (Sage)
%o R = PolynomialRing(QQ, 'x')
%o @CachedFunction
%o def skp(n, x) : # Swiss-Knife polynomials A153641.
%o if n == 0 : return 1
%o return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
%o def A162660(n,k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k]
%o matrix(ZZ, 9, A162660) # _Peter Luschny_, Jul 23 2012
%Y V_n(k), n=0, 1, ..., k=0: A155585, k=1: A009832,
%Y V_n(k), k=0, 1, ..., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536.
%Y Cf. A153641, A154341, A154342, A154343, A154344, A154345.
%K sign,tabl
%O 0,5
%A _Peter Luschny_, Jul 09 2009
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