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 A322225 Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x - k*x^2), for n >= 0. 6
 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, 40320, 109584, -204436, -699804, 442665, 1969380, -575310, -3176172, 593523, 3176172, -575310, -1969380, 442665, 699804, -204436, -109584, 40320, 362880, 1026576, -2093220, -7488928, 5124105, 24465321, -7192395, -46885278, 7343325, 57764619, -7343325, -46885278, 7192395, 24465321, -5124105, -7488928, 2093220, 1026576, -362880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS FORMULA Each row sums to 1. Left and right borders equal n! and (-1)^n*n!, respectively. EXAMPLE This irregular triangle 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; 40320, 109584, -204436, -699804, 442665, 1969380, -575310, -3176172, 593523, 3176172, -575310, -1969380, 442665, 699804, -204436, -109584, 40320; ... in which the central terms equal A322228. 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 row[n_] := CoefficientList[Product[k+x-k*x^2, {k, 1, n}] + O[x]^(2n+1), x]; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Dec 26 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("")) CROSSREFS Cf. A322226, A322227, A322228. Cf. A322235 (variant). Sequence in context: A224748 A165494 A204916 * A110049 A246577 A097032 Adjacent sequences:  A322222 A322223 A322224 * A322226 A322227 A322228 KEYWORD sign,tabf AUTHOR Paul D. Hanna, Dec 15 2018 STATUS approved

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Last modified May 18 19:48 EDT 2021. Contains 344002 sequences. (Running on oeis4.)