A387429 Self-convolution of n-th antidiagonal of the natural number array (so named in the Comments section of A000027).

1, 12, 73, 284, 835, 2036, 4347, 8408, 15069, 25420, 40821, 62932, 93743, 135604, 191255, 263856, 357017, 474828, 621889, 803340, 1024891, 1292852, 1614163, 1996424, 2447925, 2977676, 3595437, 4311748, 5137959, 6086260, 7169711, 8402272, 9798833, 11375244

Offset

1,2

Links

Table of n, a(n) for n=1..34.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = n*(4 + 5*n^2 + 3*n^4)/12.

a(n) = 6*a(n - 1) - 15*a(n - 2) + 20*a(n - 3) - 15*a(n - 4) + 6*a(n - 5) - a(n - 6).

G.f.: x*(1 + 6*x + 16*x^2 + 6*x^3 + x^4)/(-1 + x)^6.

E.g.f.: exp(x)*x*(12 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/12. - Stefano Spezia, Oct 09 2025

Example

a(3) = (4,5,6)**(6,5,4) = (4*6 + 5*5 + 6*4) = 73.

Mathematica

t[n_, k_] := n + (n + k - 2) (n + k - 1)/2;  (* A000027 and A185787 *)
s[n_] := Sum[t[k, n - k + 1]*t[n - k + 1, k], {k, 1, n}];
r = Table[s[n], {n, 1, 40}] (* A387429 *)

Crossrefs

Cf. A000027, A185787.

Sequence in context: A120783 A103475 A024014 * A069039 A156196 A041270

Adjacent sequences: A387426 A387427 A387428 * A387430 A387431 A387432

Keyword

nonn,easy

Author

Clark Kimberling, Sep 29 2025

Status

approved