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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,        5;\n  2:       6,       27,        30;\n  3:      24,      150,       309,        210;\n  4:     120,      942,      2745,       3522,       1680;\n  5:     720,     6732,     24948,      45837,      41778,      15120;\n  6:    5040,    54324,    241956,     570339,     750816,     523620,     151200;\n  7:   40320,   490032,   2533212,    7224228,   12280257,   12447936,    6969420,    1663200;\n  8:  362880,  4894128,  28679292,   95416596,  197213049,  259394508,  212100012,   98601840,   19958400;\n  9: 3628800, 53658720, 350416584, 1326996756, 3212546238, 5157714717, 5493128724, 3743318556, 1481407920, 259459200;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+5)*x+(n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (2*n + 5)*x + (n + 4)*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (2*n + 5)*x + (n + 4)*x^2)*v[i - 1])/(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","x \u21a6 x/(x+1)"],"edges":["e-4deb1a58186d6d0a","e-d751c0e9c45b3fd6"],"text":"A130534 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A130534 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}