A047835 -id:A047835 - cp's OEIS frontend

A133708 First differences of A047835.

0, 0, 1, 69, 1694, 22932, 208152, 1413720, 7697052, 35194302, 139687119, 493127635, 1577331756, 4637757488, 12679063488, 32529562560, 78917794128, 182184724908, 402332471541, 853769650041, 1747606106554, 3462012537060, 6656436729800, 12452933493000

Offset: 1

Peter Bala, Sep 21 2007

nonn

T. D. Noe, Table of n, a(n) for n = 1..1003

Cf. A040977, A047819, A047835, A107915.

Join[{0,0,1},Differences[Table[Product[Times@@((i+Range[4,7])/(i+Range[0,3])),{i,n}],{n,0,30}]]] (* Harvey P. Dale, Aug 08 2015 *)

In terms of Vandermonde determinants, a(n) = 1/864*sum {1 <= x_1,x_2,x_3 <= n} (x_1*x_2*x_3)^2*(det V(x_1,x_2,x_3))^2 = 1/864 *sum {1 <= i,j,k <= n} (i*j*k*(i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. a(n) = (n-2)*(n-1)^2*n^3*(n+1)^3*(n+2)^2*(n+3)*(n^2+n+3)*(2n+1)/108864000.

Empirical G.f.: x^3*(x+1)*(x^8+52*x^7+658*x^6+2890*x^5+4810*x^4+2890*x^3+658*x^2+52*x+1)/(x-1)^16. [Colin Barker, Jun 06 2012]

A103905 Square array T(n,k) read by antidiagonals: number of tilings of an hexagon.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 20, 20, 4, 1, 70, 175, 50, 5, 1, 252, 1764, 980, 105, 6, 1, 924, 19404, 24696, 4116, 196, 7, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1, 48620, 34763300

Offset: 1

Ralf Stephan, Feb 22 2005

As a square array, T(n,k) = number of all k-watermelons without a wall of length n. - Steven Finch, Mar 30 2008

			Array begins:
  1,   2,     3,      4,        5,         6, ...
  1,   6,    20,     50,      105,       196, ...
  1,  20,   175,    980,     4116,     14112, ...
  1,  70,  1764,  24696,   232848,   1646568, ...
  1, 252, 19404, 731808, 16818516, 267227532, ...
  ...

Peter J. Forrester and Alex Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
Anthony J. Guttmann, Aleksandr L. Owczarek and Xavier G. Viennot, Vicious walkers and Young tableaux. I. Without walls, J. Phys. A 31 (1998) 8123-8135.
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Christian Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507 [math.CO], 2005.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 28.
Percy A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.

Rows include A002415, A047819, A047835, A047831.
Columns include A000984 and A000891.
Main diagonal is A008793.
Cf. A120258, A133112.

t[n_, k_] := Product[j!*(j + 2*n)!/(j + n)!^2, {j, 0, k - 1}]; Join[{1}, Flatten[ Table[ t[n - k , k], {n, 1, 10}, {k, 1, n}]]] (* Jean-François Alcover, May 16 2012, from 2nd formula *)

T(n, k) = [V(2n+k-1)V(k-1)V(n-1)^2]/[V(2n-1)V(n+k-1)^2], with V(n) the superfactorial numbers (A000178).
T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].
T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) ]]].
T(n, k) = Prod[i=1..k, Prod[j=n+1..2n+1, i+j]/Prod[j=0..n, i+j]]; - Paul Barry, Jun 13 2006
Conjectural formula as a sum of squares of Vandermonde determinants: T(n,k) = 1/((1!*2! ... *(n-1)!)^2*n!)* sum {1 <= x_1, ..., x_n <= k} (det V(x_1, ..., x_n))^2, where V(x_1, ..., x_n) is the Vandermonde matrix of order n. Compare with A133112. - Peter Bala, Sep 18 2007
For k >= 1, T(n,k)=det(binomial(2*n,n+i-j))1<=i,j<=k [Krattenhaller, Theorem 4].
Let H(n) = product {k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Section 4.29 with x -> 1]. Setting a = b = n and c = k gives the entries for this table. - Peter Bala, Dec 22 2011

A133815 Square array of Hankel transforms of binomial(n+k,floor((n+k)/2)), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, -1, 2, 1, 1, -1, 3, 3, 1, 1, 1, 4, -6, 6, 1, 1, 1, 5, -10, 20, 10, 1, 1, -1, 6, 15, 50, -50, 20, 1, 1, -1, 7, 21, 105, -175, 175, 35, 1, 1, 1, 8, -28, 196, 490, 980, -490, 70, 1, 1, 1, 9, -36, 336, 1176, 4116, -4116, 1764, 126, 1

Offset: 0

Paul Barry, Sep 24 2007

T(n+1,k) is the Hankel transform of binomial(n+k, floor((n+k)/2)).

Even-indexed columns count tilings of hexagons: A002415 (<2,n,2>), A047819 (<3,n,3>), A047835 (<4,n,4>), etc.

			Array begins
  1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10, ...
  1,   -1,    3,   -6,   20,  -50, ...
  1,   -1,    4,  -10,   50, -175, ...
  1,    1,    5,   15,  105,  490, ...
  1,    1,    6,   21,  196, 1176, ...
As a number triangle, T(n-k,k) gives
  1;
  1,   1;
  1,   1,   1;
  1,  -1,   2,   1;
  1,  -1,   3,   3,   1;
  1,   1,   4,  -6,   6,   1;
  1,   1,   5, -10,  20,  10,   1;
  1,  -1,   6,  15,  50, -50,  20,   1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A002415, A047819, A047835, A103905, A120247.

F:= Floor;
function t(n,k)
  if k eq 0 then return 1;
  elif k eq 1 then return (-1)^F(n/2);
  elif (k mod 2) eq 0 then return (&*[ Binomial(n+F(k/2)+j, F(k/2))/Binomial(F(k/2)+j, F(k/2)) : j in [0..F((k-2)/2)] ]);
  else return (-1)^F(n/2)*(&*[ Binomial(n+F((k+1)/2)+j, F((k+1)/2))/Binomial(F((k+1)/2)+j, F((k+1)/2)) : j in [0..F((k-3)/2)] ]);
  end if;
end function;
// [[t(n,k): k in [0..10]]: n in [0..10]];
A133815:= func< n,k | t(n-k, k) >;
[A133815(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 16 2023

T[ n_, m_] := With[{k = Quotient[m + 1, 2]}, (-1)^(Quotient[n, 2] m) Product[ Binomial[n + k + j, k] / Binomial[k + j, k], {j, 0, k - 1 - Mod[m, 2]}]];
(* Michael Somos, Apr 03 2021 *)

alias(C, binomial);
T(n,k) = if (k % 2 == 0, prod(j=0, (k-2)/2, C(n+k/2+j,k/2)/C(k/2+j,k/2)), (cos(Pi*n/2)+sin(Pi*n/2))*prod(j=0, (k-3)/2, C(n+(k+1)/2+j,(k+1)/2)/C((k+1)/2+j,(k+1)/2)));
tabl(nn) = matrix(nn, nn, n, k, round(T(n-1, k-1))); \\ Michel Marcus, Dec 10 2016

T(n, m) = my(k = (m+1)\2); (-1)^(n\2*m) * prod(j=0, k-1-m%2, binomial(n+k+j, k) / binomial(k+j, k)); /* Michael Somos, Apr 03 2021 */

def f(k): return (k+1)//2
def t(n, k): return (-1)^(k*(n//2))*product(binomial(n+f(k) +j, f(k))/binomial(f(k) +j, f(k)) for j in range(f(k-1)))
def A133815(n,k): return t(n-k, k)
flatten([[A133815(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 16 2023

T(n,k) = if(k mod 2 = 0, Product_{j=0..(k-2)/2} C(n+k/2+j,k/2) / C(k/2+j,k/2), (cos(Pi*n/2) + sin(Pi*n/2))*Product_{j=0..(k-3)/2} C(n+(k+1)/2+j,(k+1)/2)/C((k+1)/2+j,(k+1)/2)).

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A133708 First differences of A047835.

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A103905 Square array T(n,k) read by antidiagonals: number of tilings of an hexagon.

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A133815 Square array of Hankel transforms of binomial(n+k,floor((n+k)/2)), read by antidiagonals.

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