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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,        3;\n  2:       5,       15,       11;\n  3:      17,       74,      106,        50;\n  4:      74,      416,      869,       800,       274;\n  5:     394,     2694,     7316,      9869,      6616,      1764;\n  6:    2484,    19916,    66149,    116551,    114939,     60172,     13068;\n  7:   18108,   166088,   649699,   1405564,   1816816,   1403492,    600104,    109584;\n  8:  149904,  1545064,  6938112,  17734032,  28227569,  28657392,  18125232,   6530904,  1026576;\n  9: 1389456, 15877080, 80339964, 236336792, 445512222, 558203799, 464947792, 248293572, 77150448, 10628640;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n^2-2*n+1+(4*n^2-6*n+2)*x+(6*n^2-6*n+1)*x^2+(4*n^2-2*n)*x^3+n^2*x^4)*P(n-2,x) + (2*n-1+(6*n-1)*x+(6*n+1)*x^2+(2*n+1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 + (4*n^2 - 6*n + 2)*x + (6*n^2 - 6*n + 1)*x^2 + (4*n^2 - 2*n)*x^3 + n^2*x^4)*P[n - 2, x] + (2*n - 1 + (6*n - 1)*x + (6*n + 1)*x^2 + (2*n + 1)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 + (4*n^2 - 6*n + 2)*x + (6*n^2 - 6*n + 1)*x^2 + (4*n^2 - 2*n)*x^3 + n^2*x^4)*v[i - 2] + (2*n - 1 + (6*n - 1)*x + (6*n + 1)*x^2 + (2*n + 1)*x^3)*v[i - 1])/(1 + 2*x + 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-efe23a76953cf69d","steps":["D+1","x \u21a6 x/(x+1)"],"edges":["e-d9ba82797b30d77a","e-b9c1c7661b4291ac"],"text":"A130534 \u2192 D+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A130534 \u2192 D+1 \u2192 x \u21a6 x/(x+1)","chain":null}