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A130713 in the columns.","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: 1;\n  1: 3,    1;\n  2: 4,    5,     1;\n  3: 4,   14,     8,     1;\n  4: 4,   32,    38,    12,     1;\n  5: 4,   68,   146,    86,    17,     1;\n  6: 4,  140,   506,   490,   171,    23,    1;\n  7: 4,  284,  1658,  2466,  1345,   309,   30,   1;\n  8: 4,  572,  5258, 11522,  9191,  3199,  519,  38,  1;\n  9: 4, 1148, 16346, 51346, 57477, 28385, 6832, 823, 47, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 4 + 5*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[2, x_] := 4 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 4 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 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-b442c9393792f928","steps":["basis A008277"],"edges":["e-82e9acb07306af49"],"text":"A167194 \u2192 basis A008277"},"headline":"A167194 \u2192 basis A008277","chain":null}