A177747 Convolution of A008805 (triangular numbers repeated) with itself.

1, 2, 7, 12, 27, 42, 77, 112, 182, 252, 378, 504, 714, 924, 1254, 1584, 2079, 2574, 3289, 4004, 5005, 6006, 7371, 8736, 10556, 12376, 14756, 17136, 20196, 23256, 27132, 31008, 35853, 40698, 46683, 52668, 59983, 67298, 76153, 85008, 95634, 106260, 118910, 131560, 146510

Offset: 0

Author

Gary W. Adamson, Dec 17 2010

Keywords

• nonn
• easy

Examples

			As a multiplication table array:
.
1, 1, 3, 3, 6,...
1, 1, 3, 3,......
3, 3, 9,.........
3, 3,............
6,...............
.
Then taking antidiagonal sums of terms, we obtain 1, (1 + 1) = 2, (3 + 1 + 3) = 7,  (3 + 3 + 3 + 3) = 12, (6, + 3 + 9 + 3 + 6) = 27, etc.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• 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 (2,3,-8,-2,12,-2,-8,3,2,-1).

Crossrefs

Cf. A008805.

Programs

Magma

A008805:=func; [&+[A008805(i)*A008805(n-i): i in [0..n]]: n in [0..44]]; // Bruno Berselli, Mar 23 2012

Mathematica

lst = CoefficientList[ Series[1/((1 - x) (1 - x^2)^2), {x, 0, 111}], x]; t[n_, k_] := lst[[n]] lst[[k]]; f[n_] := Sum[ t[n - m + 1, m], {m, n}]; Array[f, 45] (* Robert G. Wilson v, Dec 18 2010 *)
LinearRecurrence[{2, 3, -8, -2, 12, -2, -8, 3, 2, -1}, {1, 2, 7, 12, 27, 42, 77, 112, 182, 252}, 45] (* Bruno Berselli, Mar 23 2012 *)

Formula

Square (1 + x + 3x^2 + 3x^3 + 6x^4 + 6x^5 + ...)

G.f.: 1/((x+1)^4*(x-1)^6). [Bruno Berselli, Mar 23 2012]

a(n) = (n+5)*(2*n*(n+10)*(n^2+10*n+35)+5*(2*n*(n+10)+39)*(-1)^n+573)/3840. [Bruno Berselli, Mar 23 2012]

Extensions

More terms from Robert G. Wilson v, Dec 18 2010

Definition rewritten by Bruno Berselli, Mar 23 2012