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grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:       39,        60,         16;\n  1:      193,       504,        368,         64;\n  2:      897,      3336,       4176,       1984,        256;\n  3:     4007,     19404,      34864,      28416,       9984,        1024;\n  4:    17415,    104052,     244832,     286976,     172800,       48128,       4096;\n  5:    74177,    527600,    1538208,    2366080,    2049280,      976896,     225280,      16384;\n  6:   311105, 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(16*n + 32)*x^3)*P[n - 1, x] + (12*n^2 + 48*n + 33 + (32*n^2 + 128*n + 64)*x + (16*n^2 + 64*n - 32)*x^2 - 64*x^3 - 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 36*n^2 + 58*n + 33 - (8*n + 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 39 + 60*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 108*n^2 + 236*n + 165 + (16*n^3 + 112*n^2 + 236*n + 152)*x - (32*n + 60)*x^2 - (16*n + 32)*x^3)*v[i - 1] + (12*n^2 + 48*n + 33 + (32*n^2 + 128*n + 64)*x + (16*n^2 + 64*n - 32)*x^2 - 64*x^3 - 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 36*n^2 + 58*n + 33 - (8*n + 12)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-8eb9bb3a09f0b7fd","steps":["basis A013609","D+1"],"edges":["e-125a67b606fd2347","e-3165e559392a010e"],"text":"A123519 \u2192 basis 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