A178440 Convolution square of A058187, the tetrahedral series with repeats.

1, 2, 9, 16, 44, 72, 156, 240, 450, 660, 1122, 1584, 2508, 3432, 5148, 6864, 9867, 12870, 17875, 22880, 30888, 38896, 51272, 63648, 82212, 100776, 127908, 155040, 193800, 232560, 286824, 341088, 415701, 490314, 591261, 692208, 826804, 961400, 1138500, 1315600

Offset: 0

Author

Gary W. Adamson, Dec 22 2010

Keywords

• nonn
• easy

Examples

			Antidiagonal sums of terms in the array:
  1,.. 1,. 4,. 4,. 10,  10,...
  1,.. 1,. 4,. 4,. 10,........
  4,.. 4,.16,.16,.............
  4,.. 4,.16,.................
  10,.10,.....................
  10,.........................
Example: a(4) = 44 = (10 + 4 + 16 + 4 + 10).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
• Index entries for linear recurrences with constant coefficients, signature (2, 5, -12, -9, 30, 5, -40, 5, 30, -9, -12, 5, 2, -1).

Crossrefs

Cf. A058187.

Programs

Mathematica

LinearRecurrence[{2,5,-12,-9,30,5,-40,5,30,-9,-12,5,2,-1},{1,2,9,16,44,72,156,240,450,660,1122,1584,2508,3432},40] (* Harvey P. Dale, Apr 17 2020 *)

Formula

Square (1 + x + 4x^2 + 4x^3 + 10x^4 + ...) = (1 + 2x + 9x^2 + ...).

G.f.: 1 / ( (1+x)^6*(x-1)^8 ). - R. J. Mathar, Jul 21 2015

Extensions

Corrected by R. J. Mathar, Jul 21 2015