A110517 -id:A110517 - cp's OEIS frontend

A109466 Riordan array (1, x(1-x)).

1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 1, -3, 1, 0, 0, 0, 3, -4, 1, 0, 0, 0, -1, 6, -5, 1, 0, 0, 0, 0, -4, 10, -6, 1, 0, 0, 0, 0, 1, -10, 15, -7, 1, 0, 0, 0, 0, 0, 5, -20, 21, -8, 1, 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1, 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1, 0, 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Aug 28 2005

Inverse is Riordan array (1, xc(x)) (A106566).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008
Coefficient array of the polynomials Chebyshev_U(n, sqrt(x)/2)*(sqrt(x))^n. - Paul Barry, Sep 28 2009

			Rows begin:
  1;
  0,  1;
  0, -1,  1;
  0,  0, -2,  1;
  0,  0,  1, -3,  1;
  0,  0,  0,  3, -4,   1;
  0,  0,  0, -1,  6,  -5,   1;
  0,  0,  0,  0, -4,  10,  -6,   1;
  0,  0,  0,  0,  1, -10,  15,  -7,  1;
  0,  0,  0,  0,  0,   5, -20,  21, -8,  1;
  0,  0,  0,  0,  0,  -1,  15, -35, 28, -9, 1;
From _Paul Barry_, Sep 28 2009: (Start)
Production array is
  0,    1,
  0,   -1,    1,
  0,   -1,   -1,   1,
  0,   -2,   -1,  -1,   1,
  0,   -5,   -2,  -1,  -1,  1,
  0,  -14,   -5,  -2,  -1, -1,  1,
  0,  -42,  -14,  -5,  -2, -1, -1,  1,
  0, -132,  -42, -14,  -5, -2, -1, -1,  1,
  0, -429, -132, -42, -14, -5, -2, -1, -1, 1 (End)

Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A026729 (unsigned version), A000108, A030528, A124644.

/* As triangle */ [[(-1)^(n-k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 14 2016

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-#)&, 13] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Number triangle T(n, k) = (-1)^(n-k)*binomial(k, n-k).
T(n, k)*2^(n-k) = A110509(n, k); T(n, k)*3^(n-k) = A110517(n, k).
Sum_{k=0..n} T(n,k)*A000108(k)=1. - Philippe Deléham, Jun 11 2007
From Philippe Deléham, Oct 30 2008: (Start)
Sum_{k=0..n} T(n,k)*A144706(k) = A082505(n+1).
Sum_{k=0..n} T(n,k)*A002450(k) = A100335(n).
Sum_{k=0..n} T(n,k)*A001906(k) = A100334(n).
Sum_{k=0..n} T(n,k)*A015565(k) = A099322(n).
Sum_{k=0..n} T(n,k)*A003462(k) = A106233(n). (End)
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = -12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. - Philippe Deléham, Oct 27 2008
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 28 2008
G.f.: 1/(1-y*x+y*x^2). - Philippe Deléham, Dec 15 2011
T(n,k) = T(n-1,k-1) - T(n-2,k-1), T(n,0) = 0^n. - Philippe Deléham, Feb 15 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = F(n+1,-x) where F(n,x)is the n-th Fibonacci polynomial in x defined in A011973. - Philippe Deléham, Feb 22 2013
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 26 2013
Sum_{k=0..n} T(n,k)*T(n+1,k) = -A110320(n). - Philippe Deléham, Feb 26 2013
For T(0,0) = 0, the signed triangle below has the o.g.f. G(x,t) = [t*x(1-x)]/[1-t*x(1-x)] = L[t*Cinv(x)] where L(x) = x/(1-x) and Cinv(x)=x(1-x) with the inverses Linv(x) = x/(1+x) and C(x)= [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108, so the inverse o.g.f. is Ginv(x,t) = C[Linv(x)/t] = [1-sqrt[1-4*x/(t(1+x))]]/2 (cf. A124644 and A030528). - Tom Copeland, Jan 19 2016

A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 29 2005

A skew version of Sierpinski’s triangle A047999. - Johannes W. Meijer, Jun 05 2011
Row sums are A002487(n+1). Diagonal sums are A106345. Inverse is A106346.
Triangle formed by reading T triangle mod 2 with T := A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167. - Philippe Deléham, Dec 18 2008

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 0, 1;

G. C. Greubel, Rows n = 0..50, flattened
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

Cf. A047999, A002487.
Cf. A106345 (diagonal sums), A106346 (inverse).
Cf. A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167.

Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # G. C. Greubel, Feb 07 2020

[ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 07 2020

seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # G. C. Greubel, Feb 07 2020

Table[Mod[Binomial[k, n-k], 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 18 2017 *)

T(n,k) = binomial(k,n-k)%2;
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

[[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Feb 07 2020

A110518 Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 18, 6, 1, 0, 135, 45, 9, 1, 0, 1134, 378, 81, 12, 1, 0, 10206, 3402, 756, 126, 15, 1, 0, 96228, 32076, 7290, 1296, 180, 18, 1, 0, 938223, 312741, 72171, 13365, 2025, 243, 21, 1, 0, 9382230, 3127410, 729729, 138996, 22275, 2970, 315, 24, 1, 0

Offset: 0

Paul Barry, Jul 24 2005

Row sums are C(3;n), A064063. Inverse is A110517. Diagonal sums are A110525.

			Rows begin
  1;
  0,    1;
  0,    3,    1;
  0,   18,    6,    1;
  0,  135,   45,    9,    1;
  0, 1134,  378,   81,   12,    1;
  ...
Production matrix begins:
  0,   1;
  0,   3,   1;
  0,   9,   3,   1;
  0,  27,   9,   3,   1;
  0,  81,  27,   9,   3,   1;
  0, 243,  81,  27,   9,   3,   1;
  ... - _Philippe Deléham_, Sep 23 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := (k/n)*3^(n - k)*Binomial[2*n - k - 1, n - k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

concat([1], for(n=1,10, for(k=0,n, print1((k/n)*3^(n-k)*binomial(2*n-k-1,n-k), ", ")))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(0,k) = 0^k, T(n,k) = (k/n)*C(2n-k-1, n-k)*3^(n-k), n > 0, k > 0.
T(n,k) = A106566(n,k)*3^(n-k). - Philippe Deléham, Nov 08 2007
Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 3, 3, 3, 3, 3, 3, 3, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 23 2014

A110519 Riordan array (1/(1-xc(3x)), xc(3x)/(1-xc(3x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 25, 33, 9, 1, 190, 256, 78, 13, 1, 1606, 2186, 703, 139, 17, 1, 14506, 19863, 6591, 1430, 216, 21, 1, 137089, 188449, 63813, 14669, 2501, 309, 25, 1, 1338790, 1845416, 633808, 151532, 27940, 3980, 418, 29, 1, 13403950, 18513822, 6425196, 1580316, 307752, 48180, 5931, 543, 33, 1

Offset: 0

Paul Barry, Jul 24 2005

Product of (1, x*c(3*x)) and (1/(1-x), x/(1-x)) (A110518 and A007318). The binomial transform of the inverse of this triangle has general element (-3)^(n-k)*C(k,n-k), that is, it is the Riordan array (1, x*(1-3*x)) [A110517]. Row sums are A110520. Diagonal sums are A110521.

			Rows begin
     1;
     1,    1;
     4,    5,    1;
    25,   33,    9,    1;
   190,  256,   78,   13,    1;
  1606, 2186,  703,  139,   17,    1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := Sum[j*3^(n - j)*Binomial[2*n - j - 1, n - j]*Binomial[j, k]/n, {j, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

concat([1], for(n=1, 10, for(k=0,n, print1(sum(j=0,n, j*binomial(2*n-j-1,n-j)*binomial(j,k)*3^(n-j)/n), ", ")))) \\ G. C. Greubel, Aug 29 2017

Number triangle T(0,k) = 0^k, T(n,k) = Sum_{j=0..n} j*C(2n-j-1, n-j)* C(j, k)3^(n-j)/n, n > 0, k > 0. Deleham triangle Delta(0^n, 3-2*0^n) [see construction in A084938].

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

Original entry on oeis.org

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)

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A109466 Riordan array (1, x(1-x)).

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A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

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A110518 Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

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A110519 Riordan array (1/(1-x*c(3*x)), x*c(3*x)/(1-x*c(3*x))), c(x) the g.f. of A000108.

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

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A110519 Riordan array (1/(1-xc(3x)), xc(3x)/(1-xc(3x))), c(x) the g.f. of A000108.

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