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T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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:       2,       12;\n  2:       6,       60,        81;\n  3:      24,      312,       729,        432;\n  4:     120,     1848,      5751,       6048,       2025;\n  5:     720,    12528,     46980,      66960,      40500,       8748;\n  6:    5040,    96336,    413424,     719280,     597375,     236196,     35721;\n  7:   40320,   831168,   3957660,    7959168,    8150625,    4470228,   1250235,    139968;\n  8:  362880,  7964352,  41229324,   92740032,  110664225,   75442752,  29505546,   6158592,   531441;\n  9: 3628800, 83998080, 466052616, 1147290048, 1541361150, 1232496972, 603113364, 177199488, 28697814, 1968300;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+12*n*x-81*x^2)*P(n-2,x) + (12*n*x^2+36*x^3)*d/dx P(n-2,x) + (2*n+1+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 12*n*x - 81*x^2)*P[n - 2, x] + (12*n*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 12*n*x - 81*x^2)*v[i - 2] + (12*n*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); 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-efe23a76953cf69d","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-4deb1a58186d6d0a","e-96bcb9de644eb07a"],"text":"A130534 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A130534 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}