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 A213547 Antidiagonal sums of the convolution array A213505. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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). 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 EXAMPLE 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) MATHEMATICA (See A213505.) PROG (PARI) {a(n) = n++; (n^6 - n^2) / 60}; /* Michael Somos, Oct 08 2017 */ CROSSREFS Cf. A000290, A000578, A213500, A213505. Sequence in context: A289384 A200205 A059585 * A050484 A359715 A238536 Adjacent sequences: A213544 A213545 A213546 * A213548 A213549 A213550 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jun 16 2012 STATUS approved

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