A213547 Antidiagonal sums of the convolution array A213505.

1, 12, 68, 260, 777, 1960, 4368, 8856, 16665, 29524, 49764, 80444, 125489, 189840, 279616, 402288, 566865, 784092, 1066660, 1429428, 1889657, 2467256, 3185040, 4069000, 5148585, 6456996, 8031492, 9913708, 12149985, 14791712, 17895680, 21524448, 25746721, 30637740

Offset: 1

Author

Clark Kimberling, Jun 16 2012

Keywords

• nonn
• easy

Comments

Also, the antidiagonal sums of the convolution array A213555.

An m-star is an m-antichain with a smallest element adjoined. Then, a(n) is the number of proper mergings of a 2-star and an (n-1)-chain, see example. - Henri Mühle, Jan 23 2013

Convolution of A000290 and A000578. - Stefano Spezia, Apr 07 2023

Examples

			From _Henri Mühle_, Jan 23 2013: (Start)
For n=2, let S=({s0,s1,s2},{(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2)}) be a 2-star, and let C=({c},{(c,c)}) be a 1-chain. The a(2)=12 proper mergings of S and C are:
({s0,s1,s2,c},{(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s0),(c,s1),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s1),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s2),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s1),(c,s2),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(c,s1),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(c,s1),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s1,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s2,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s1,c),(s2,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
(End)
		

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 (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000290, A000578, A213500, A213505.

Programs

Mathematica

(See A213505.)

PARI

{a(n) = n++; (n^6 - n^2) / 60}; /* Michael Somos, Oct 08 2017 */

Formula

a(n) = (n^6 + 6*n^5 + 15*n^4 + 20*n^3 + 14*n^2 + 4*n)/60.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^7.

a(n) = a(-2-n) and a(n-1) = (n^6 - n^2) / 60 for all n in Z. - Michael Somos, Oct 08 2017

E.g.f.: exp(x)*x*(60 + 300*x + 350*x^2 + 140*x^3 + 21*x^4 + x^5)/60. - Stefano Spezia, Apr 07 2023