A213560 Antidiagonal sums of the convolution array A213558.

1, 24, 236, 1400, 6009, 20608, 59952, 153792, 357225, 765688, 1535820, 2913560, 5270993, 9153600, 15339712, 24914112, 39357873, 60656664, 91429900, 135083256, 195987209, 279684416, 393128880, 544960000, 745814745, 1008681336, 1349297964, 1786600216, 2343221025

Offset: 1

Author

Clark Kimberling, Jun 17 2012

Keywords

• nonn
• easy

Comments

An m-star is an m-antichain with a smallest element adjoined. Then, a(n) is the number of proper mergings of a 3-star and an (n-1)-chain. - Henri Mühle, Jan 23 2013

Convolution of A000578 and A000583. - Stefano Spezia, Apr 07 2023

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Henri Muehle, Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details, arXiv preprint arXiv:1301.1654 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000578, A000583, A213558, A213500.

Programs

Mathematica

(See A213558.)

Formula

a(n) = n*(1 + n)^2*(2 + n)*(16 + 18*n + 21*n^2 + 12*n^3 + 3*n^4)/840.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).

G.f.: x*(1 + x)*(1 + 4*x + x^2)*(1 + 10*x + x^2)/(1 - x)^9.