A080851 Square array of pyramidal numbers, read by antidiagonals.

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78

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.

Cf. A057145, A027660 (antidiagonal sums).

See A257199 for another version of this array.

Sequence in context: A193043 A086271 A345229 * A209518 A108285 A207619

Adjacent sequences: A080848 A080849 A080850 * A080852 A080853 A080854

Keyword

nonn,tabl,easy

Author

Paul Barry, Feb 21 2003

Status

approved