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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

LINKS

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

P. Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.

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.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

G.f.: (1-xy)/(1-2xy-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[0](x) = 1

p[1](x) = x

p[2](x) = 1 + x^2

p[3](x) = 3*x + x^3

p[4](x) = 1 + 6*x^2 + x^4

p[5](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, x) :

    return 1 if n==0 else add(binomial(n, k)*(x^(n-k)) for k in range(n+1)[::2])

def A119467_row(n) :

    R = PolynomialRing(ZZ, 'x')

    return R(A119467_poly(n, x)).coeffs()

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[0]: 1,1,1,1,1,1, ....... A000012

p[1]: 0,1,2,3,4,5, ....... A001477

p[2]: 1,2,5,10,17,26, .... A002522

p[3]: 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 January 18 01:05 EST 2020. Contains 330995 sequences. (Running on oeis4.)