A275490 Square array of 5D pyramidal numbers, read by antidiagonals.

1, 1, 5, 1, 6, 15, 1, 7, 21, 35, 1, 8, 27, 56, 70, 1, 9, 33, 77, 126, 126, 1, 10, 39, 98, 182, 252, 210, 1, 11, 45, 119, 238, 378, 462, 330, 1, 12, 51, 140, 294, 504, 714, 792, 495, 1, 13, 57, 161, 350, 630, 966, 1254, 1287, 715, 1, 14, 63, 182, 406, 756, 1218, 1716, 2079, 2002, 1001

Offset: 2

Author

R. J. Mathar, Jul 30 2016

Keywords

• nonn
• easy
• tabl

Comments

Let F(r,n,d) = binomial(r+d-2,d-1)* ((r-1)*(n-2)+d) /d be the d-dimensional pyramidal numbers. Then A(n,k) = F(k,n,5).

Sum of the n-th antidiagonal is binomial(n+4,7) + binomial(n+4,5) = A055797(n-1). - Mathew Englander, Oct 27 2020

Examples

			The array starts in rows n>=2 and columns k>=1 as
   1    5   15   35   70  126  210  330  495   715  1001  1365  1820
   1    6   21   56  126  252  462  792 1287  2002  3003  4368  6188
   1    7   27   77  182  378  714 1254 2079  3289  5005  7371 10556
   1    8   33   98  238  504  966 1716 2871  4576  7007 10374 14924
   1    9   39  119  294  630 1218 2178 3663  5863  9009 13377 19292
   1   10   45  140  350  756 1470 2640 4455  7150 11011 16380 23660
   1   11   51  161  406  882 1722 3102 5247  8437 13013 19383 28028
   1   12   57  182  462 1008 1974 3564 6039  9724 15015 22386 32396
   1   13   63  203  518 1134 2226 4026 6831 11011 17017 25389 36764
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150, flattened)
• Index to sequences related to polygonal numbers

Crossrefs

Cf. Row sums of A080852 (4D), A080851 (3D), A057145 (2D), A077028 (1D).

Cf. A055797.

Programs

Mathematica

Table[Binomial[k + 3, 4] + (# - 2)*Binomial[k + 3, 5] &[m - k + 1], {m, 2, 12}, {k, m - 1}] // Flatten (* Michael De Vlieger, Nov 05 2020 *)

Formula

A(n+2,k) = Sum_{j=0..k-1} A080852(n,j).

A(n,k) = binomial(k+3,4) + (n-2)*binomial(k+3,5). - Mathew Englander, Oct 27 2020