OFFSET
0,3
COMMENTS
The first row contains the tetrahedral numbers, which are really three-dimensional, but can be regarded as degenerate 4D pyramidal numbers. - N. J. A. Sloane, Aug 28 2015
FORMULA
T(n, k) = binomial(k + 4, 4) + (n-1)*binomial(k + 3, 4), corrected Oct 01 2021.
T(n, k) = T(n - 1, k) + C(k + 3, 4) = T(n - 1, k) + k(k + 1)(k + 2)(k + 3)/24.
G.f. for rows: (1 + nx)/(1 - x)^5, n >= -1.
T(n,k) = sum_{j=0..k} A080851(n,j). - R. J. Mathar, Jul 28 2016
EXAMPLE
Array, n >= 0, k >= 0, begins
1 4 10 20 35 56 ...
1 5 15 35 70 126 ...
1 6 20 50 105 196 ...
1 7 25 65 140 266 ...
1 8 30 80 175 336 ...
MAPLE
A080852 := proc(n, k)
binomial(k+4, 4)+(n-1)*binomial(k+3, 4) ;
end proc:
seq( seq(A080852(d-k, k), k=0..d), d=0..12) ; # R. J. Mathar, Oct 01 2021
MATHEMATICA
T[n_, k_] := Binomial[k+3, 3] + Binomial[k+3, 4]n;
Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)
PROG
(Derive) vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^5, x, 11), x, n), n, 0, 11), k, -1, 10)
(Derive) VECTOR(VECTOR(comb(k+3, 3)+comb(k+3, 4)n, k, 0, 11), n, 0, 11)
CROSSREFS
See A257200 for another version of the array.
KEYWORD
AUTHOR
Paul Barry, Feb 21 2003
STATUS
approved