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 A119467 A masked Pascal triangle. 10
 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 1, 0, 15, 0, 15, 0, 1, 0, 7, 0, 35, 0, 21, 0, 1, 1, 0, 28, 0, 70, 0, 28, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 1, 0, 45, 0, 210, 0, 210, 0, 45, 0, 1, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 1, 0, 66, 0, 495, 0, 924 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x). Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry, May 26 2006 Rows give coefficients of polynomials p_n(x) = Sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k) having e.g.f. exp(x*t)*cosh(t)= 1*(t^0/0!) + x*(t^1/1!) + (1+x^2)*(t^2/2!) + ... - Peter Luschny, Jul 14 2009 Inverse of the coefficient matrix of the Swiss-Knife polynomials in ascending order of x^i (reversed and aerated rows of A153641). - Peter Luschny, Jul 16 2012 Call this array M and 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 matrix product M(0)*M(1)*M(2)*... is equal to A136630 but with the first row and column omitted. - Peter Bala, Jul 28 2014 The row polynomials SKv(n,x) = [(x+1)^n + (x-1)^n]/2 , with e.g.f. cosh(t)*exp(xt), are the umbral compositional inverses of the row polynomials of A119879 (basically the Swiss Knife polynomials SK(n,x) of A153641); i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119879 are an inverse pair. Both sequences of polynomials are Appell sequences, i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SKv(.,0)+x)^n = SKv(n,x), reflecting that the first column has the e.g.f. cosh(t). The raising operator is R = x + tanh(d/dx); i.e., R SKv(n,x) = SKv(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of cosh(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015 The signed version of this matrix has the e.g.f. cos(t) e^{xt}, generating Appell polynomials that have only real, simple zeros and whose extrema are maxima above the x-axis and minima below and situated above and below the zeros of the next lower degree polynomial. The bivariate versions appear on p. 27 of Dimitrov and Rusev in conditions for entire functions that are cosine transforms of a class of functions to have only real zeros. - Tom Copeland, May 21 2020 LINKS Reinhard Zumkeller, Rows n = 0..125 of table, flattened Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28. Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021. T. Copeland, Skipping over Dimensions, Juggling Zeros in the Matrix, 2020. D. Dimitrov and P. Rusev, Zeros of entire Fourier transforms, East Journal on Approximations, Vol. 17, No. 1, p. 1-108, 2011. Miguel Méndez, Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO}, 2017, Section 4.3, Example 4. Miguel A. Méndez, Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25. FORMULA G.f.: (1-x*y)/(1-2*x*y-x^2+x^2*y^2); T(n,k) = C(n,k)*(1+(-1)^(n-k))/2; Column k has g.f. (1/(1-x^2)(x/(1-x^2))^k*Sum_{j=0..k+1, binomial(k+1,j)*sin((j+1)*Pi/ 2)^2*x^j}. Column k has e.g.f. cosh(x)*x^k/k!. - Paul Barry, May 26 2006 Let Pascal's triangle, A007318 = P; then this triangle = (1/2) * (P + 1/P). Also A131047 = (1/2) * (P - 1/P). - Gary W. Adamson, Jun 12 2007 Equals A007318 - A131047 since the zeros of the triangle are masks for the terms of A131047. Thus A119467 + A131047 = Pascal's triangle. - Gary W. Adamson, Jun 12 2007 T(n,k) = (A007318(n,k) + A130595(n,k))/2, 0<=k<=n. - Reinhard Zumkeller, Mar 23 2014 EXAMPLE Triangle begins   1,   0, 1,   1, 0,  1,   0, 3,  0,  1,   1, 0,  6,  0,   1,   0, 5,  0, 10,   0,   1,   1, 0, 15,  0,  15,   0,   1,   0, 7,  0, 35,   0,  21,   0,  1,   1, 0, 28,  0,  70,   0,  28,  0,  1,   0, 9,  0, 84,   0, 126,   0, 36,  0, 1,   1, 0, 45,  0, 210,   0, 210,  0, 45, 0, 1 p(x) = 1 p(x) = x p(x) = 1 + x^2 p(x) = 3*x + x^3 p(x) = 1 + 6*x^2 + x^4 p(x) = 5*x + 10*x^3 + x^5 Connection with A136630: With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins /1        \/1        \/1        \      /1         \ |0 1      ||0 1      ||0 1      |      |0 1       | |1 0 1    ||0 0 1    ||0 0 1    |... = |1 0  1    | |0 3 0 1  ||0 1 0 1  ||0 0 0 1  |      |0 4  0 1  | |1 0 6 0 1||0 0 3 0 1||0 0 1 0 1|      |1 0 10 0 1| |...      ||...      ||...      |      |...       | - Peter Bala, Jul 28 2014 MAPLE # Polynomials: p_n(x) p := proc(n, x) local k, pow; pow := (n, k) -> `if`(n=0 and k=0, 1, n^k); add((k+1 mod 2)*binomial(n, k)*pow(x, n-k), k=0..n) end; # Coefficients: a(n) seq(print(seq(coeff(i!*coeff(series(exp(x*t)*cosh(t), t, 16), t, i), x, n), n=0..i)), i=0..8); # Peter Luschny, Jul 14 2009 MATHEMATICA Table[Binomial[n, k] (1 + (-1)^(n - k))/2, {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 06 2015 *) PROG (Sage) @CachedFunction def A119467_poly(n):     R = PolynomialRing(ZZ, 'x')     x = R.gen()     return R.one() if n==0 else R.sum(binomial(n, k)*x^(n-k) for k in range(0, n+1, 2)) def A119467_row(n):     return list(A119467_poly(n)) for n in (0..10) : print(A119467_row(n)) # Peter Luschny, Jul 16 2012 (Haskell) a119467 n k = a119467_tabl !! n !! k a119467_row n = a119467_tabl !! n a119467_tabl = map (map (flip div 2)) \$                zipWith (zipWith (+)) a007318_tabl a130595_tabl -- Reinhard Zumkeller, Mar 23 2014 (MAGMA) /* As triangle */ [[Binomial(n, k)*(1 + (-1)^(n - k))/2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 26 2015 CROSSREFS Cf. A131047, A153641. From Peter Luschny, Jul 14 2009: (Start) Cf. A034839, A162590. p[n](k), n=0,1,... k= 0: 1,  0,   1,    0,    1,   0, ... A128174 k= 1: 1,  1,   2,    4,    8,  16, ... A011782 k= 2: 1,  2,   5,   14,   41, 122, ... A007051 k= 3: 1,  3,  10,   36,  136,      ... A007582 k= 4: 1,  4,  17,   76,  353,      ... A081186 k= 5: 1,  5,  26,  140,  776,      ... A081187 k= 6: 1,  6,  37,  234, 1513,      ... A081188 k= 7: 1,  7,  50,  364, 2696,      ... A081189 k= 8: 1,  8,  65,  536, 4481,      ... A081190 k= 9: 1,  9,  82,  756, 7048,      ... A060531 k=10: 1, 10, 101, 1030,            ... A081192 p[n](k), k=0,1,... p: 1,1,1,1,1,1, ....... A000012 p: 0,1,2,3,4,5, ....... A001477 p: 1,2,5,10,17,26, .... A002522 p: 0,4,14,36,76,140, .. A079908 (End) Cf. A000182, A133314, A153641. Sequence in context: A112743 A230427 A229995 * A166353 A110235 A036856 Adjacent sequences:  A119464 A119465 A119466 * A119468 A119469 A119470 KEYWORD easy,nonn,tabl,look AUTHOR Paul Barry, May 21 2006 EXTENSIONS Edited by N. J. A. Sloane, Jul 14 2009 STATUS approved

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Last modified October 26 00:52 EDT 2021. Contains 348256 sequences. (Running on oeis4.)