A104969 -id:A104969 - cp's OEIS frontend

A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.

1, -1, 1, -1, -2, 1, -1, -1, -3, 1, -1, 0, 0, -4, 1, -1, 1, 2, 2, -5, 1, -1, 2, 3, 4, 5, -6, 1, -1, 3, 3, 3, 5, 9, -7, 1, -1, 4, 2, 0, 0, 4, 14, -8, 1, -1, 5, 0, -4, -6, -6, 0, 20, -9, 1, -1, 6, -3, -8, -10, -12, -14, -8, 27, -10, 1, -1, 7, -7, -11, -10, -10, -14, -22, -21, 35, -11, 1, -1, 8, -12, -12, -5, 0, 0, -8, -27, -40, 44, -12, 1

Offset: 0

Paul D. Hanna, Mar 30 2005

Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.

Riordan array ((1-2*x)/(1-x), x(1-2*x)/(1-x)). - Philippe Deléham, Dec 05 2015

			Triangle begins:
   1;
  -1,  1;
  -1, -2,  1;
  -1, -1, -3,  1;
  -1,  0,  0, -4,  1;
  -1,  1,  2,  2, -5,  1;
  -1,  2,  3,  4,  5, -6,  1;
  -1,  3,  3,  3,  5,  9, -7,  1;
  -1,  4,  2,  0,  0,  4, 14, -8,  1;
  -1,  5,  0, -4, -6, -6,  0, 20, -9, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080

Cf. A078011, A090132, A104968, A104969, A134824, A153881.

Cf. A347171 (rows reversed, up to signs).

A104967:= func< n,k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;
[A104967(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 09 2021

A104967:= (n,k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);
seq(seq( A104967(n,k), k=0..n), n=0..12); # G. C. Greubel, Jun 09 2021

T[n_, k_]:= T[n, k]= Which[k==n, 1, k==0, 0, True, T[n-1, k-1] - Sum[T[n-i, k-1], {i, 2, n-k+1}]];
Table[T[n, k], {n, 13}, {k, n}]//Flatten (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

T(n,k):=sum((-2)^i*binomial(k+1,i)*binomial(n-i,k),i,0,n-k); /* Vladimir Kruchinin, Nov 02 2011 */

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)),n,x),k,y)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

def A104967_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A104967_row(n)) # Peter Luschny, Mar 16 2016

G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).
Sum_{k=0..n} abs(T(n, k)) = A104968(n).
Sum_{k=0..n} T(n, k)^2 = A104969(n).
T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - Vladimir Kruchinin, Nov 02 2011
Sum_{k=0..floor(n/2)} T(n-k, k) = A078011(n+2). - G. C. Greubel, Jun 09 2021

A104968 Absolute row sums of triangle A104967.

Original entry on oeis.org

1, 2, 4, 6, 6, 12, 22, 32, 34, 52, 100, 150, 170, 266, 438, 640, 766, 1196, 1996, 2888, 3210, 4994, 8534, 12392, 15106, 22154, 34366, 52134, 62148, 96956, 156396, 217416, 262062, 394164, 643908, 950944, 1150368, 1689176, 2600992, 3767888, 4840338

Offset: 0

Paul D. Hanna, Mar 30 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A104967, A104969.

A104967[n_, k_]:= A104967[n, k]= Sum[(-2)^j*Binomial[k+1, j]*Binomial[n-j, k], {j, 0, n-k}];
A104968[n_]:= A104968[n]= Sum[Abs[A104967[n, k]], {k,0,n}];
Table[A104968[n], {n, 0, 50}] (* G. C. Greubel, Jun 09 2021 *)

{a(n)=local(X=x+x*O(x^n)); sum(k=0,n,abs(polcoeff(polcoeff((1-2*X)/(1-X-X*y*(1-2*X)),n,x),k,y)))}

@cached_function
def A104967(n,k): return sum( (-2)^j*binomial(k+1,j)*binomial(n-j,k) for j in (0..n-k))
def A104968(n): return sum( abs(A104967(n,k)) for k in (0..n))
[A104968(n) for n in (0..50)] # G. C. Greubel, Jun 09 2021

a(n) = Sum_{k=0..n} abs(A104967(n,k)).

A104970 Sum of squares of terms in even-indexed rows of triangle A104967.

Original entry on oeis.org

1, 6, 18, 92, 298, 1444, 4852, 22840, 78490, 362580, 1265564, 5767688, 20366596, 91866984, 327351336, 1464522864, 5257011066, 23361650484, 84371466636, 372831130344, 1353477992556, 5952169844664, 21704580414936, 95051752387344

Offset: 0

Paul D. Hanna, Mar 30 2005

nonn

Sum of squares of terms in odd-indexed rows of triangle A104967 equals twice this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A104967, A104968, A104969.

A104970:= func< n | n eq 0 select 1 else  4^n + (&+[(-1)^j*2^(2*n-2*j-1)*Binomial(2*j+1,j+1): j in [0..n-1]]) >;
[A104970(n): n in [0..40]]; // G. C. Greubel, Jun 09 2021

Flatten[{1,Table[2^(2*n-1)*(2+Sum[(-1)^k*Binomial[2*k+1,k+1]/2^(2*k),{k,0,n-1}]),{n,1,20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

{a(n)=local(X=x+x*O(x^(2*n))); sum(k=0,2*n,polcoeff(polcoeff((1-2*X)/(1-X-X*y*(1-2*X)),2*n,x),k,y)^2)}

@cached_function
def A104967(n,k): return sum( (-2)^j*binomial(k+1,j)*binomial(n-j,k) for j in (0..n-k))
def A104970(n): return sum((A104967(2*n,k))^2 for k in (0..2*n))
[A104970(n) for n in (0..50)] # G. C. Greubel, Jun 09 2021

G.f. A(x) satisfies: 2*(1+12*x)*A(x) - (1-16*x^2)*deriv(A(x), x) + 4 = 0.

a(n) = 2^(2*n-1)*(2 + Sum_{k=0..n-1} (-1)^k*binomial(2*k+1,k+1)/2^(2*k)). - Vaclav Kotesovec, Oct 28 2012

A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

Original entry on oeis.org

1, 1, 1, 1, 2, -1, 1, 3, -1, 1, 1, 4, 0, 0, -1, 1, 5, 2, -2, 1, 1, 1, 6, 5, -4, 3, -2, -1, 1, 7, 9, -5, 3, -3, 3, 1, 1, 8, 14, -4, 0, 0, 2, -4, -1, 1, 9, 20, 0, -6, 6, -4, 0, 5, 1, 1, 10, 27, 8, -14, 12, -10, 8, -3, -6, -1, 1, 11, 35, 21, -22, 14, -10, 10, -11, 7, 7, 1

Offset: 0

Kevin Ryde, Aug 21 2021

Doche and Mendès France form polynomials P_n(y) = Sum_{j=0..2^n-1} GRS(j) * y^wt(j) and here row n is the coefficients of P_n starting from the constant term, so P_n(y) = Sum_{k=0..n} T(n,k)*y^k. They conjecture that the number of real roots of P_n is A285869(n).

Row sum n is the sum of GRS terms from j = 0 to 2^n-1 inclusive, which Brillhart and Morton (Beispiel 6 page 129) show is A020986(2^n-1) = 2^ceiling(n/2) = A060546(n). The same follows by substituting y=1 in the P_n recurrence or the generating function.

			Triangle begins
        k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=0:   1
  n=1:   1,  1
  n=2:   1,  2, -1
  n=3:   1,  3, -1,  1
  n=4:   1,  4,  0,  0, -1
  n=5:   1,  5,  2, -2,  1,  1
  n=6:   1,  6,  5, -4,  3, -2, -1
  n=7:   1,  7,  9, -5,  3, -3,  3,  1
For T(5,3), those j in the range 0 <= j < 2^5 with wt(j) = 3 are
  j      =  7 11 13 14 19 21 22 25 26 28
  GRS(j) = +1 -1 -1 +1 -1 +1 -1 -1 -1 +1 total -2 = T(5,3)

Kevin Ryde, Table of n, a(n) for rows 0 to 100, flattened
John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, Illinois Journal of Mathematics, volume 22, issue 1, 1978, pages 126-148.
Christophe Doche and Michel Mendès France, An Exercise on the Average Number of Real Zeros of Random Real Polynomials, Finite and Infinite Combinatorics conference, Budapest, 2001, pages 1-14, see Rudin-Shapiro example page 9.

Cf. A020985 (GRS), A020986 (GRS partial sums), A000120 (binary weight), A285869.
Columns k=0..3: A000012, A001477, A000096, A275874.
Cf. A165326 (main diagonal), A248157 (second diagonal negated).
Cf. A060546 (row sums), A104969 (row sums squared terms).
Cf. A329301 (antidiagonal sums).
Cf. A104967 (rows reversed, up to signs).

my(M=Mod('x, 'x^2-(1-'y)*'x-2*'y)); row(n) = Vecrev(subst(lift(M^n),'x,'y+1));

T(n,k) = T(n-1,k) - T(n-1,k-1) + 2*T(n-2,k-1) for n>=2, and taking T(n,k)=0 if k<0 or k>n.
T(n,k) = (-1)^k * A104967(n,n-k).
Row polynomial P_n(y) = (1-y)*P_{n-1}(y) + 2*y*P_{n-2}(y) for n>=2. [Doche and Mendès France]
G.f.: (1 + 2*x*y)/(1 + x*(y-1) - 2*x^2*y).
Column g.f.: C_k(x) = 1/(1-x) for k=0 and C_k(x) = x^k * (2*x-1)^(k-1) / (1-x)^(k+1) for k>=1.

cp's OEIS Frontend

A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A104968 Absolute row sums of triangle A104967.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A104970 Sum of squares of terms in even-indexed rows of triangle A104967.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula