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 A322893 a(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1. 4
 1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, 23818189395, 907365113622, 24884202594186, 1097379059482797, 35843982129214455, 1794829778206820280, 68106808437178597960, 3815489686616468849025, 165072679883587905823683, 10226191400763164277215330, 497092886801366317217274750, 33732223801436694239674078341, 1820835126778068312737993859263 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..300 FORMULA a(n) = A322891(n, n-1) for n >= 1. a(n) = A322891(n, n+1)/2 for n >= 1. a(n) = n*(n+1)/2 * A322894(n) for n >= 1. EXAMPLE The irregular triangle A322891 of coefficients of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins 1; 1, 1, 2; 2, 3, 9, 6, 8; 6, 11, 42, 45, 84, 44, 48; 24, 50, 227, 310, 717, 620, 908, 400, 384; 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840; 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ... Note that this sequence forms a secondary diagonal in the above triangle [1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...] and may be divided by triangular numbers n*(n+1)/2 to obtain A322894: [1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...]. PROG (PARI) {A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)} /* Print the irregular triangle */ for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print("")) /* Print this sequence */ for(n=1, 30, print1( A322891(n, n-1), ", ")) CROSSREFS Cf. A322891, A322892, A322894. Cf. A322237 (variant). Sequence in context: A229071 A220857 A157537 * A114943 A119577 A051273 Adjacent sequences:  A322890 A322891 A322892 * A322894 A322895 A322896 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 29 2018 STATUS approved

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Last modified July 22 05:47 EDT 2019. Contains 325213 sequences. (Running on oeis4.)