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A213547 Antidiagonal sums of the convolution array A213505.

%I A213547 #41 Apr 07 2023 11:27:20
%S A213547 1,12,68,260,777,1960,4368,8856,16665,29524,49764,80444,125489,189840,
%T A213547 279616,402288,566865,784092,1066660,1429428,1889657,2467256,3185040,
%U A213547 4069000,5148585,6456996,8031492,9913708,12149985,14791712,17895680,21524448,25746721,30637740
%N A213547 Antidiagonal sums of the convolution array A213505.
%C A213547 Also, the antidiagonal sums of the convolution array A213555.
%C A213547 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
%C A213547 Convolution of A000290 and A000578. - _Stefano Spezia_, Apr 07 2023
%H A213547 Clark Kimberling, <a href="/A213547/b213547.txt">Table of n, a(n) for n = 1..1000</a>
%H A213547 Henri Muehle, <a href="http://arxiv.org/abs/1301.1654">Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details</a>, arXiv preprint arXiv:1301.1654 [math.CO], 2013.
%H A213547 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A213547 a(n) = (n^6 + 6*n^5 + 15*n^4 + 20*n^3 + 14*n^2 + 4*n)/60.
%F A213547 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).
%F A213547 G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^7.
%F A213547 a(n) = a(-2-n) and a(n-1) = (n^6 - n^2) / 60 for all n in Z. - _Michael Somos_, Oct 08 2017
%F A213547 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
%e A213547 From _Henri Mühle_, Jan 23 2013: (Start)
%e A213547 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:
%e A213547 ({s0,s1,s2,c},{(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(c,s0),(c,s1),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(c,s1),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(c,s2),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(c,s1),(c,s2),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s0,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s0,c),(c,s1),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s0,c),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s0,c),(c,s1),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s1,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s2,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 ({s0,s1,s2,c},{(s1,c),(s2,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
%e A213547 (End)
%t A213547 (See A213505.)
%o A213547 (PARI) {a(n) = n++; (n^6 - n^2) / 60}; /* _Michael Somos_, Oct 08 2017 */
%Y A213547 Cf. A000290, A000578, A213500, A213505.
%K A213547 nonn,easy
%O A213547 1,2
%A A213547 _Clark Kimberling_, Jun 16 2012