

A080852


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


14



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, 1
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OFFSET

0,3


COMMENTS

The first row contains the tetrahedral numbers, which are really threedimensional, 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..66.
Index to sequences related to pyramidal numbers


FORMULA

T(n, k) = C(k + 3, 3) + n*C(k + 3, 4).
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 ...


PROG

(Derive) vector(vector(poly_coeff(Taylor((1+kx)/(1x)^5, x, 11), x, n), n, 0, 11), k, 1, 10) 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.
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



