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A080852 Square array of 4D pyramidal numbers, read by antidiagonals. 12
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 (list; table; graph; refs; listen; history; text; internal format)
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..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)/(1-x)^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

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Last modified February 19 08:50 EST 2018. Contains 299330 sequences. (Running on oeis4.)