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A178301 Triangle T(n,k) = binomial(n,k)*binomial(n+k+1,n+1) read by rows, 0<=k<=n. 4
1, 1, 3, 1, 8, 10, 1, 15, 45, 35, 1, 24, 126, 224, 126, 1, 35, 280, 840, 1050, 462, 1, 48, 540, 2400, 4950, 4752, 1716, 1, 63, 945, 5775, 17325, 27027, 21021, 6435, 1, 80, 1540, 12320, 50050, 112112, 140140, 91520, 24310, 1, 99, 2376, 24024, 126126, 378378, 672672, 700128, 393822, 92378 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Antidiagonal sums are given by A113682. - Johannes W. Meijer, Mar 24 2013

LINKS

Table of n, a(n) for n=0..54.

Author?, Norm of a continuous function, dxdy.ru (in Russian)

FORMULA

T(n,k) = A007318(n,k) * A178300(n+1,k+1).

From Peter Bala, Jun 18 2015: (Start)

n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k+1,n+1)*x^k = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+1,k+1)*binomial(n+k+1,n+1)*(1 + x)^k.

Recurrence: (2*n - 1)*(n + 1)*R(n,x) = 2*(4*n^2*x + 2*n^2 - x - 1)*R(n-1,x) - (2*n + 1)(n - 1)*R(n-2,x) with R(0,x) = 1, R(1,x) = 1 + 3*x.

A182626(n) = -R(n-1,-2) for n >= 1. (End)

From Peter Bala, Jul 20 2015: (Start)

n-th row polynomial R(n,x) = Jacobi_P(n,0,1,2*x + 1).

(1 + x)*R(n,x) gives the row polynomials of A123160.

(End)

G.f.: (1+x-sqrt(1-2*x+x^2-4*x*y))/(2*(1+y)*x*sqrt(1-2*x+x^2-4*x*y)). - Emanuele Munarini, Dec 16 2016

R(n,x) = Sum_{k=0..n} (-1)^k*(2k+1)*P(k,2k+1)/(n+1), where P(k,x) is the k-th Legendre polynomial (A100258). - Max Alekseyev, Jun 28 2018

Polynomial g(n,x) = R(n,-x)/(n+1) delivers the maximum of f(1)^2/(Integral_0^1 f(x)^2 dx) over all polynomials f(x) with real coefficients and deg(f(x))<=n. This maximum equals (n+1)^2. See dxdy.ru link. - Max Alekseyev, Jun 28 2018

EXAMPLE

n=0: 1;

n=1: 1,3;

n=2: 1,8,10;

n=3: 1,15,45,35;

n=4: 1,24,126,224,126;

n=5: 1,35,280,840,1050,462;

n=6: 1,48,540,2400,4950,4752,1716;

n=7: 1,63,945,5775,17325,27027,21021,6435;

MAPLE

A178301 := proc(n, k)

        binomial(n, k)*binomial(n+k+1, n+1) ;

end proc: # R. J. Mathar, Mar 24 2013

MATHEMATICA

Flatten[Table[Binomial[n, k]Binomial[n+k+1, n+1], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Aug 23 2014 *)

PROG

(Maxima) create_list(binomial(n, k)*binomial(n+k+1, n+1), n, 0, 12, k, 0, n); Emanuele Munarini, Dec 16 2016

CROSSREFS

Cf. A007318, A047781 (row sums), A178300, A182626, A123160.

Sequence in context: A008298 A039692 A071815 * A120236 A049760 A019146

Adjacent sequences:  A178298 A178299 A178300 * A178302 A178303 A178304

KEYWORD

easy,nonn,tabl

AUTHOR

Alford Arnold, May 30 2010

STATUS

approved

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Last modified April 11 18:59 EDT 2021. Contains 342888 sequences. (Running on oeis4.)