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 A322228 a(n) = [x^n] Product_{k=1..n} (k + x - k*x^2), for n >= 0. 4
 1, 1, -3, -21, 75, 1475, -5005, -221389, 593523, 57764619, -89101881, -23273632371, 953636541, 13409519997705, 23908442020749, -10469975115603501, -40844292735050541, 10646036726696597027, 66995992524016223543, -13672657170891872702719, -122282221141986787179519, 21647316686778755963070321, 256325163531592225309743129, -41426918732532942751217361155, -620418821801458605268716606275, 94275566307675915918535250768725 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n+1) = -2*(n+1) * A322227(n) + a(n), for n >= 1. a(n+1) = -n*(n+1)^2 * A322226(n) + a(n), for n >= 1. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 EXAMPLE The irregular triangle A322225 formed from coefficients of x^k in Product_{m=1..n} (m + x - m*x^2), for n >= 0, k = 0..2*n, begins 1; 1, 1, -1; 2, 3, -3, -3, 2; 6, 11, -12, -21, 12, 11, -6; 24, 50, -61, -140, 75, 140, -61, -50, 24; 120, 274, -375, -1011, 540, 1475, -540, -1011, 375, 274, -120; 720, 1764, -2696, -8085, 4479, 15456, -5005, -15456, 4479, 8085, -2696, -1764, 720; 5040, 13068, -22148, -71639, 42140, 169266, -50932, -221389, 50932, 169266, -42140, -71639, 22148, 13068, -5040; ... in which the central terms equal this sequence. RELATED SEQUENCES. Note that the terms in the secondary diagonal A322227 in the above triangle [1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...] may be divided by triangular numbers to obtain A322226: [1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...]. MATHEMATICA a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n}]; Array[a, 26, 0] (* Jean-François Alcover, Dec 29 2018 *) PROG (PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x - m*x^2) +x*O(x^k), k)} /* Print the irregular triangle */ for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print("")) /* Print this sequence */ for(n=0, 30, print1( T(n, n), ", ")) CROSSREFS Cf. A322235, A322236, A322237. Cf. A322238 (variant). Sequence in context: A238193 A054646 A228317 * A109721 A067002 A110450 Adjacent sequences:  A322225 A322226 A322227 * A322229 A322230 A322231 KEYWORD sign AUTHOR Paul D. Hanna, Dec 15 2018 STATUS approved

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Last modified June 18 23:16 EDT 2021. Contains 345125 sequences. (Running on oeis4.)