A110524 -id:A110524 - cp's OEIS frontend

A110522 Riordan array (1/(1+x), x(1-2x)/(1+x)^2).

1, -1, 1, 1, -5, 1, -1, 12, -9, 1, 1, -22, 39, -13, 1, -1, 35, -115, 82, -17, 1, 1, -51, 270, -344, 141, -21, 1, -1, 70, -546, 1106, -773, 216, -25, 1, 1, -92, 994, -2954, 3199, -1466, 307, -29, 1, -1, 117, -1674, 6888, -10791, 7461, -2487, 414, -33, 1, 1, -145, 2655, -14484, 31179, -30645, 15060, -3900, 537, -37, 1

Offset: 0

Paul Barry, Jul 24 2005

Inverse of A110519.

Product of inverse binomial transform matrix (1/(1+x), x/(1+x)) and (1, x*(1-3*x)) (A110517).

			Rows begin
   1;
  -1,    1;
   1,   -5,    1;
  -1,   12,   -9,    1;
   1,  -22,   39,  -13,    1;
  -1,   35, -115,   82,  -17,    1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, J. Int. Seq. 16 (2013) #13.5.4.

Cf. A110519 (inverse), A110523 (row sums), A110524 (diagonal sums).

Cf. A000326, A016813, A033999, A052924, A078005, A110517, A236267.

A110522:= func< n,k | (-1)^(n+k)*(&+[ 3^(j-k)*Binomial(k,j-k)*Binomial(n,j) : j in [0..n]] ) >;
[A110522(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 28 2023

T[n_,k_]:= Sum[(-1)^(n-j)*(-3)^(j-k)*Binomial[k, j- k]*Binomial[n, j], {j,0,n}];
Table[T[n,k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 30 2017 *)

A110522(n,k) = if(n==0, 1, sum(j=0,n, (-1)^(n-j)*(-3)^(j-k)*binomial(n,j)*binomial(k, j-k)));
for(n=0,12, for(k=0,n, print1(A110522(n,k), ", "))) \\ G. C. Greubel, Aug 30 2017; Dec 28 2023

def A110522(n,k): return (-1)^(n+k)*sum(3^(j-k)*binomial(k,j-k)*binomial(n,j) for j in range(n+1))
flatten([[A110522(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 28 2023

Number triangle T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k)*2^i.
Sum_{k=0..n} T(n, k) = A110523(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A110524(n) (diagonal sums).
T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - 2*T(n-2,k-1), T(0,0) = 1, T(1,0) = -1, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014
From G. C. Greubel, Dec 28 2023: (Start)
T(n, 0) = A033999(n).
T(n, 1) = (-1)^(n-1)*A000326(n), n >= 1.
T(n, n) = 1.
T(n, n-1) = -A016813(n-1), n >= 1.
T(n, n-2) = A236267(n-2), n >= 2.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A052924(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A078005(n). (End)

A077999 Expansion of (1-x)/(1-2x-2x^3).

Original entry on oeis.org

1, 1, 2, 6, 14, 32, 76, 180, 424, 1000, 2360, 5568, 13136, 30992, 73120, 172512, 407008, 960256, 2265536, 5345088, 12610688, 29752448, 70195072, 165611520, 390727936, 921846016, 2174915072, 5131286016, 12106264064, 28562358272, 67387288576, 158987105280

Offset: 0

N. J. A. Sloane, Nov 17 2002

a(n) = number of permutations on [n] that avoid nonconsecutive instances of the patterns 321 and 312. For example, a(4) does not count pi=4231 because 431 forms a 321 pattern in pi but 431 is not a consecutive (that is, contiguous) string in pi; also, the first 3 letters form a 312 pattern but that's not disqualifying because they do occur consecutively. Counting these permutations by various statistics yields the listed formulas/recurrences. - David Callan, Oct 26 2006

a(n) = term (1,1) of M^n, M = the 4 X 4 matrix [1,0,1,1; 1,1,0,0; 0,1,0,1; 1,0,0,1]. a(n)/a(n-1) tends to 2.3593040859..., an eigenvalue of the matrix and a root to the characteristic polynomial x^4 - 3x^3 + 2x^2 - 2x + 2. - Gary W. Adamson, Oct 01 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. R. Finch, Several Constants Arising in Statistical Mechanics, arXiv:math/9810155 [math.CO], 1998-1999. see p. 8.
Index entries for linear recurrences with constant coefficients, signature (2,0,2)

Cf. A052912, A110524.

a:=[1,1,2];; for n in [4..40] do a[n]:=2*(a[n-1]+a[n-3]); od; a; # G. C. Greubel, Jun 27 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/( 1-2*x-2*x^3) )); // G. C. Greubel, Jun 27 2019

CoefficientList[Series[(1-x)/(1-2x-2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,2},{1,1,2},40] (* Harvey P. Dale, Sep 10 2016 *)

my(x='x+O('x^40)); Vec((1-x)/(1-2*x-2*x^3)) \\ G. C. Greubel, Jun 27 2019

((1-x)/(1-2*x-2*x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(n) = 2*(a(n-1) + a(n-3)) counts the above permutations by first entry. a(n) = a(n-1) + a(n-2) + 3*Sum_{k=0..n-3} a(k) counts by last entry. a(n) = 2^(n-1) + Sum_{k=0..n-3} 2^(n-2-k)*a(k) counts by location of first 3xx pattern. a(n) = Sum_{k=0..floor(n/3)} ((n-k)/(n-2k))* binomial(n-2*k,k) * 2^(n-2*k-1) counts by number of 3xx patterns. - David Callan, Oct 26 2006
a(n) = A052912(n) - A052912(n-1). - R. J. Mathar, May 30 2014
a(n) = (-1)^n * A110524(n). - G. C. Greubel, Jun 27 2019

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A110522 Riordan array (1/(1+x), x(1-2x)/(1+x)^2).

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A077999 Expansion of (1-x)/(1-2x-2x^3).

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cp's OEIS Frontend

A110522 Riordan array (1/(1+x), x*(1-2*x)/(1+x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A077999 Expansion of (1-x)/(1-2*x-2*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A110522 Riordan array (1/(1+x), x(1-2x)/(1+x)^2).

A077999 Expansion of (1-x)/(1-2x-2x^3).