A076035 -id:A076035 - cp's OEIS frontend

A067347 Square array read by antidiagonals: T(n,k)=(T(n,k-1)n^2-Catalan(k-1)n)/(n-1) with a(n,0)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 14, 20, 12, 4, 1, 0, 42, 70, 51, 20, 5, 1, 0, 132, 252, 222, 104, 30, 6, 1, 0, 429, 924, 978, 548, 185, 42, 7, 1, 0, 1430, 3432, 4338, 2904, 1150, 300, 56, 8, 1, 0, 4862, 12870, 19323, 15432, 7170, 2154, 455, 72, 9, 1, 0, 16796

Offset: 0

Henry Bottomley, Jan 16 2002

			Array begins
1 0 0 0 0 0 0 0 ... k=0
1 1 2 5 14 42 132 429 ... k=1
1 2 6 20 70 252 924 3432 ... k=2
1 3 12 51 222 978 4338 19323 ... k=3

Rows give A000007, A000108, A000984, A007854, A076035, A076036. Columns give A000012, A001700, A002378, A062158.

T(n, k) =A067345(n, k)*n =A067346(n, k)*n/(n-1)

A127543 Triangle T(n,k), 0<=k<=n, read by rows given by :[ -1,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, -1, 1, -1, 1, -2, 0, 2, -1, 1, -6, 2, 1, 3, -1, 1, -18, 5, 7, 2, 4, -1, 1, -57, 17, 19, 13, 3, 5, -1, 1, -186, 56, 64, 36, 20, 4, 6, -1, 1, -622, 190, 212, 124, 56, 28, 5, 7, -1, 1, -2120, 654, 722, 416, 198, 79, 37, 6, 8, -1, 1, -7338, 2282, 2494, 1434, 673, 287, 105, 47, 7, 9, -1, 1

Offset: 0

Philippe Deléham, Apr 01 2007

Riordan array (2/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x))). - Philippe Deléham, Jan 27 2014

			Triangle begins:
    1;
   -1,  1;
    0, -1,  1;
   -1,  1, -1,  1;
   -2,  0,  2, -1,  1;
   -6,  2,  1,  3, -1,  1;
  -18,  5,  7,  2,  4, -1,  1;
  -57, 17, 19, 13,  3,  5, -1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

A065600[n_, k_]:= If[k==n, 1, Sum[j*Binomial[k+j, j]*Binomial[2*(n-k-j), n-k]/(n-k-j), {j,0, Floor[(n-k)/2]}]];
A127543[n_, k_]:= A065600[n-1,k-1] - A065600[n-1,k];
Table[A127543[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 17 2021 *)

def A065600(n,k): return 1 if (k==n) else sum( j*binomial(k+j, j)*binomial(2*(n-k-j), n-k)/(n-k-j) for j in (0..(n-k)//2) )
def A127543(n,k): return A065600(n-1, k-1) - A065600(n-1, k)
flatten([[A127543(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 17 2021

T(n,k) = A065600(n-1,k-1) - A065600(n-1,k).
Sum_{k=0..n} T(n,k)*x^k = A127053(n), A126985(n), A127016(n), A127017(n), A126987(n), A126986(n), A126982(n), A126984(n), A126983(n), A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for n= -8,-7,...,8,9 respectively.
Sum_{j>=0} T(n,j)*A007318(j,k) = A106566(n,k).
Sum_{j>=0} T(n,j)*A038207(j,k) = A039599(n,k).
Sum_{j>=0} T(n,j)*A027465(j,k) = A116395(n,k).

A117380 Riordan array (1/(1-4xc(x)),xc(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 4, 1, 20, 5, 1, 104, 26, 6, 1, 548, 137, 33, 7, 1, 2904, 726, 178, 41, 8, 1, 15432, 3858, 954, 228, 50, 9, 1, 82128, 20532, 5100, 1242, 288, 60, 10, 1, 437444, 109361, 27233, 6701, 1601, 359, 71, 11, 1, 2331128, 582782, 145338, 35977, 8744, 2043, 442, 83, 12

Offset: 0

Paul Barry, Mar 10 2006

Triangle factors as (1,xc(x))*(1/(1-4x),x). Inverse of A117377. First row is A076035. Second row is A076025(n)-0^n. Row sums are A076025(n+1). Diagonal sums are A117381.

			Triangle begins
1,
4, 1,
20, 5, 1,
104, 26, 6, 1,
548, 137, 33, 7, 1,
2904, 726, 178, 41, 8, 1
Production array begins
4, 1
4, 1, 1
4, 1, 1, 1
4, 1, 1, 1, 1
4, 1, 1, 1, 1, 1
4, 1, 1, 1, 1, 1, 1
4, 1, 1, 1, 1, 1, 1, 1
4, 1, 1, 1, 1, 1, 1, 1, 1
... - _Philippe Deléham_, Mar 05 2013

Number triangle T(0,0)=1, T(n,k)=[k<=n]*sum{j=0..n, (j/(n-j))*C(2n-j,n-j)[k<=j]*4^(j-k)}

A133443 a(n) = Sum_{k=0..n} C(n,floor(k/2))(-1)^k3^(n-k).

Original entry on oeis.org

1, 2, 8, 24, 84, 272, 920, 3040, 10180, 33840, 112968, 376224, 1254696, 4181088, 13939248, 46459584, 154873860, 516229040, 1720795880, 5735921440, 19119861304, 63732624672, 212442552528, 708140901184, 2360471473384, 7868234639072, 26227455730640

Offset: 0

Philippe Deléham, Nov 26 2007, Dec 07 2007

nonn

Hankel transform is 4^n. Second binomial transform is A076035.

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

Cf. A000108, A015518, A053121, A076035, A127362.

Table[Sum[Binomial[n,Floor[k/2]]*(-1)^k*3^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = Sum_{k=0..n} A053121(n,k)*A015518(k+1) = (-1)^n*A127362(n). G.f.: (1/sqrt(1-4*x^2))*(1-x*c(x^2))/(1-3*x*c(x^2)), where c(x) is the g.f. of Catalan numbers A000108.
Recurrence: 3*n*a(n) = 2*(5*n-3)*a(n-1) + 4*(3*n-1)*a(n-2) - 40*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2*10^n/3^(n+1). - Vaclav Kotesovec, Oct 20 2012

More terms from Vincenzo Librandi, May 25 2013

cp's OEIS Frontend

A067347 Square array read by antidiagonals: T(n,k)=(T(n,k-1)n^2-Catalan(k-1)n)/(n-1) with a(n,0)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

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A127543 Triangle T(n,k), 0<=k<=n, read by rows given by :[ -1,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A117380 Riordan array (1/(1-4xc(x)),xc(x)), c(x) the g.f. of A000108.

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Formula

A133443 a(n) = Sum_{k=0..n} C(n,floor(k/2))(-1)^k3^(n-k).

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cp's OEIS Frontend

A067347 Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1)*n)/(n-1) with a(n,0)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

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Author

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Formula

A127543 Triangle T(n,k), 0<=k<=n, read by rows given by :[ -1,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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Author

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Comments

Examples

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Programs

Mathematica

Sage

Formula

A117380 Riordan array (1/(1-4*x*c(x)),xc(x)), c(x) the g.f. of A000108.

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Author

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Comments

Examples

Formula

A133443 a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-1)^k*3^(n-k).

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Programs

Mathematica

Formula

Extensions

A067347 Square array read by antidiagonals: T(n,k)=(T(n,k-1)n^2-Catalan(k-1)n)/(n-1) with a(n,0)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

A117380 Riordan array (1/(1-4xc(x)),xc(x)), c(x) the g.f. of A000108.

A133443 a(n) = Sum_{k=0..n} C(n,floor(k/2))(-1)^k3^(n-k).