A067001 Triangle T(n,k) = d(n-k,n), 0 <= k <= n, where d(l,m) = Sum_{k=l..m} 2^k * binomial(2*m-2*k, m-k) * binomial(m+k, m) * binomial(k, l).

1, 4, 6, 24, 60, 42, 160, 560, 688, 308, 1120, 5040, 8760, 7080, 2310, 8064, 44352, 99456, 114576, 68712, 17556, 59136, 384384, 1055040, 1572480, 1351840, 642824, 134596, 439296, 3294720, 10695168, 19536000, 21778560, 14912064, 5864640, 1038312

Offset: 0

Author

N. J. A. Sloane, Feb 16 2002

Keywords

• nonn
• tabl

Comments

For an explanation on how this triangular array is related to the Boros-Moll polynomial P_n(x) and the theory in Comtet (1967), see my comments in A223549. For example, the bivariate o.g.f. below follows from the theory in Comtet (1967). - Petros Hadjicostas, May 24 2020

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) starts:
     1;
     4,    6;
    24,   60,   42;
   160,  560,  688,  308;
  1120, 5040, 8760, 7080, 2310;
  ...
		

Links

• Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
• George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
• William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
• William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
• Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87.
• V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
• V. H. Moll, Combinatorial sequences arising from a rational integral, Onl. J. Anal. Combin. no 2 (2007) #4.

Crossrefs

Column k=0 gives A059304.

Row sums give A002458.

Main diagonal gives A004982.

Cf. A223549, A223550.

Programs

Maple

d := proc(l,m) local k; add(2^k*binomial(2*m-2*k,m-k)*binomial(m+k,m)*binomial(k,l),k=l..m); end:
T:= (n, k)-> d(n-k, n):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

T[n_, k_] := SeriesCoefficient[Sqrt[(1+y)/(1 - 8x (1+y))/(1 + y Sqrt[1 - 8x (1+y)])], {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 05 2020 *)

PARI

d(l, m) = sum(kk=l, m, 2^kk*binomial(2*m-2*kk,m-kk)*binomial(m+kk,m)*binomial(kk,l));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(d(n-k, n), ", ");); print(););} \\ Michel Marcus, Jul 18 2015

Formula

From Petros Hadjicostas, May 24 2020: (Start)

T(n,k) = 2^(2*n)*A223549(n,n-k)/A223550(n,n-k).

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = sqrt((1 + y)/(1 - 8*x*(1 + y))/(1 + y*sqrt(1 - 8*x*(1 + y)))). (End)