A076025 -id:A076025 - cp's OEIS frontend

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

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)}

A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)xC)/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 5, 1, 1, 4, 10, 14, 14, 1, 1, 5, 17, 35, 42, 42, 1, 1, 6, 26, 74, 126, 132, 132, 1, 1, 7, 37, 137, 326, 462, 429, 429, 1, 1, 8, 50, 230, 726, 1446, 1716, 1430, 1430, 1, 1, 9, 65, 359, 1434, 3858, 6441, 6435, 4862, 4862, 1, 1, 10, 82

Offset: 0

N. J. A. Sloane, Oct 29 2002

			Array begins
1 1 1 2 5 14 42 ... (n=0)
1 1 2 5 14 42 132 ... (n=1)
1 1 3 10 35 126 ... (n=2)
1 1 4 17 74 326 ...

Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076038, A067347.

C(x)=(1/2-1/2*(1-4*x)^(1/2))/x; D(x)=(1-(m-1)*x*C(x))/(1-m*x*C(x)); for(i=0,15, forstep(m=i,0,-1,print1(polcoeff(D(x),i-m),","));print()) (Klasen)

More terms from Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 12 2005

A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 10, 14, 14, 1, 5, 17, 35, 42, 42, 1, 6, 26, 74, 126, 132, 132, 1, 7, 37, 137, 326, 462, 429, 429, 1, 8, 50, 230, 726, 1446, 1716, 1430, 1430, 1, 9, 65, 359, 1434, 3858, 6441, 6435, 4862, 4862, 1, 10, 82, 530, 2582, 8952, 20532, 28770, 24310, 16796, 16796

Offset: 0

N. J. A. Sloane, Oct 29 2002

			Array begins as:
  1 1  2  5  14  42 ... (n=0)
  1 2  5 14  42 132 ... (n=1)
  1 3 10 35 126 ... (n=2)
  1 4 17 74 326 ...
  ...

Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076037, A067347.

Cf. A059718.

Unprotect[Power]; Power[0,0]=1; Protect[Power]; A[n_, m_]:= 1/(m+1)*Sum[Binomial[2*m-k, m]*(k+1)*(n-m)^k,{k,0,m}]; Table[A[n,m],{n,0,10},{m,0,n}]//Flatten (* Stefano Spezia, Sep 01 2025 *)

A(n, m) = 1/(m+1)*Sum_{k=0..m} binomial(2*m-k, m)*(k+1)*(n-m)^k, m=0..n.

More terms from Vladeta Jovovic, Jul 18 2003

a(63)-a(65) from Stefano Spezia, Sep 01 2025

A104531 Expansion of (1+sqrt(1-4x))/(5sqrt(1-4*x)-3).

Original entry on oeis.org

1, 4, 24, 148, 920, 5736, 35808, 223668, 1397496, 8732920, 54575888, 341082504, 2131706864, 13322959888, 83267756400, 520420803060, 3252620324280, 20328841669080, 127055130786960, 794094089779800, 4963086293860560, 31019282772508080, 193870492861908480

Offset: 0

Paul Barry, Mar 12 2005

Apply the Riordan matrix ((1+sqrt(1-4x))/2,(1-sqrt(1-4x))/2) (inverse of (1/(1-x),x(1-x))) to 5^n.

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

Cf. A088218, A067336, A076025, A104530, A104532.

CoefficientList[Series[(1+Sqrt[1-4*x])/(5*Sqrt[1-4*x]-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec((1+sqrt(1-4*x))/(5*sqrt(1-4*x)-3)) \\ Joerg Arndt, May 13 2013

a(n) = 0^n + sum{k=0..n, 4^(k+1)*C(2n-1, n-k-1)*2*(k+1)/(n+k+1)}
D-finite with recurrence: 4*n*a(n) = (41*n-24)*a(n-1) - 50*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3*5^(2*n-1)/4^n. - Vaclav Kotesovec, Oct 17 2012

cp's OEIS Frontend

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

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A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)xC)/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

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A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

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A104531 Expansion of (1+sqrt(1-4x))/(5sqrt(1-4*x)-3).

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cp's OEIS Frontend

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

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Examples

Formula

A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)*x*C)/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

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Extensions

A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

Views

Author

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Examples

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Programs

Mathematica

Formula

Extensions

A104531 Expansion of (1+sqrt(1-4*x))/(5*sqrt(1-4*x)-3).

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Author

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Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

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

A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)xC)/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

A104531 Expansion of (1+sqrt(1-4x))/(5sqrt(1-4*x)-3).