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

0,3

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

Links

Table of n, a(n) for n=0..65.

Index to sequences related to pyramidal numbers

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

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.

Sequence in context: A155060 A153426 A261720 * A204201 A090842 A120868

Adjacent sequences: A080849 A080850 A080851 * A080853 A080854 A080855

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 21 2003

Status

approved