OFFSET
0,3
COMMENTS
The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016
EXAMPLE
Array begins (n>=0, k>=0):
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... A000217
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... A000292
1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ... A000330
1, 6, 18, 40, 75, 126, 196, 288, 405, 550, ... A002411
1, 7, 22, 50, 95, 161, 252, 372, 525, 715, ... A002412
1, 8, 26, 60, 115, 196, 308, 456, 645, 880, ... A002413
1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
MAPLE
A080851 := proc(n, k)
binomial(k+3, 3)+(n-1)*binomial(k+2, 3) ;
end proc:
seq( seq(A080851(d-k, k), k=0..d), d=0..12) ; # R. J. Mathar, Oct 01 2021
MATHEMATICA
pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)
PROG
(Derive) vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4, x, 11), x, n), n, 0, 11), k, -1, 10) VECTOR(VECTOR(comb(k+2, 2)+comb(k+2, 3)n, k, 0, 11), n, 0, 11)
CROSSREFS
Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
See A257199 for another version of this array.
KEYWORD
AUTHOR
Paul Barry, Feb 21 2003
STATUS
approved