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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 The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial binomial(x+n,n)*binomial(x+n,n-1) in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022 Chapoton's observation above is correct: the precise expansion is binomial(x+n,n)*binomial(x+n,n-1) = Sum_{k = 0..n-1} (-1)^k*T(n-1,n-1-k)*binomial(x+2*n-1-k,2*n-1-k), as can be verified using the WZ algorithm. For example, n = 4 gives binomial(x+4,4)*binomial(x+4,3) = 35*binomial(x+7,7) - 45*binomial(x+6,6) + 15*binomial(x+5,5) - binomial(x+4,4). - Peter Bala, Jun 24 2023 LINKS David A. Corneth, Table of n, a(n) for n = 0..10010 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)^(n+k)*(2*k+1)*P(k,2*x+1)/(n+1), where P(k,x) is the k-th Legendre polynomial (cf. A100258) and P(k,2*x+1) is the k-th shifted Legendre polynomial (cf. A063007). - Max Alekseyev, Jun 28 2018; corrected by Peter Bala, Aug 08 2021 Polynomial g(n,x) = R(n,-x)/(n+1) delivers the maximum of f(1)^2/(Integral_{x=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 R := proc(n) add((-1)^(n+k)*(2*k+1)*orthopoly:-P(k, 2*x+1)/(n+1), k=0..n) end: for n from 0 to 6 do seq(coeff(R(n), x, k), k=0..n) od; # Peter Luschny, Aug 25 2021 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 (PARI) R(n, x) = sum(k=0, n, (-1)^(n+k) * (2*k+1) * pollegendre(k, 2*x+1)) / (n+1); \\ Max Alekseyev, Aug 25 2021 CROSSREFS Cf. A007318, A047781 (row sums), A178300, A182626, A123160, A132813. Sequence in context: A008298 A039692 A071815 * A120236 A049760 A343812 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 October 4 22:44 EDT 2023. Contains 365888 sequences. (Running on oeis4.)