A080852 Square array of 4D pyramidal numbers, read by antidiagonals.

1, 1, 4, 1, 5, 10, 1, 6, 15, 20, 1, 7, 20, 35, 35, 1, 8, 25, 50, 70, 56, 1, 9, 30, 65, 105, 126, 84, 1, 10, 35, 80, 140, 196, 210, 120, 1, 11, 40, 95, 175, 266, 336, 330, 165, 1, 12, 45, 110, 210, 336, 462, 540, 495, 220, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 286

Offset: 0

Author

Paul Barry, Feb 21 2003

Keywords

• easy
• nonn
• tabl

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

Examples

			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 ...
		

Links

• Index to sequences related to pyramidal numbers

Crossrefs

Rows include A000292, A000332, A002415, A001296, A002418, A002419, A051740, A051797.

Cf. A057145, A080851, A180266, A055796 (antidiagonal sums).

See A257200 for another version of the array.

Programs

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)

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 *)

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