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 A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials. 7
 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, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0, 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0, 0, 79360, 0, -32640, 0, 4032, 0, -240, 0, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 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. (1) V_n(0) = 2^n * Euler(n,1) for n > 0, A155585. (2) V_n(1) = 1 - Euler(n). (3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642. (4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1. (5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586. 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. 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 P_{F}[n](x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n. V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials, W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials. B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials, E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials. 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. 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 LINKS Table of n, a(n) for n=0..65. Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008. Peter Luschny, The Swiss-Knife polynomials. Peter Luschny, Swiss-Knife Polynomials and Euler Numbers. Wikipedia, Bernoulli number. J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232. FORMULA 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 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!) + ... V_n(x) = -x^n + Sum_{k=0..n} C(n,k)*Euler(k)*(x+1)^(n-k). EXAMPLE Triangle begins: [0] 0; [1] 1, 0; [2] 0, 2, 0; [3] -2, 0, 3, 0; [4] 0, -8, 0, 4, 0; [5] 16, 0, -20, 0, 5, 0; [6] 0, 96, 0, -40, 0, 6, 0; [7] -272, 0, 336, 0, -70, 0, 7, 0; [8] 0, -2176, 0, 896, 0, -112, 0, 8, 0; [9] 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0; MAPLE # Polynomials V_n(x): 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: # Coefficients a(n): seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t), t, 16), t, n), x, k), k=0..n)), n=0..8); MATHEMATICA 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 *) PROG (Sage) R = PolynomialRing(QQ, 'x') @CachedFunction def skp(n, x) : # Swiss-Knife polynomials A153641. if n == 0 : return 1 return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2]) def A162660(n, k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k] matrix(ZZ, 9, A162660) # Peter Luschny, Jul 23 2012 CROSSREFS V_n(k), n=0, 1, ..., k=0: A155585, k=1: A009832, V_n(k), k=0, 1, ..., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536. Cf. A153641, A154341, A154342, A154343, A154344, A154345. Sequence in context: A278520 A239246 A171700 * A324848 A090330 A332447 Adjacent sequences: A162657 A162658 A162659 * A162661 A162662 A162663 KEYWORD sign,tabl AUTHOR Peter Luschny, Jul 09 2009 STATUS approved

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Last modified April 25 03:15 EDT 2024. Contains 371964 sequences. (Running on oeis4.)