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Triangle read by rows, for n >= 0 and 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:       1;\n  1:       2,        4;\n  2:       6,       22,        20;\n  3:      24,      124,       212,       120;\n  4:     120,      788,      1928,      2084,        840;\n  5:     720,     5688,     17872,     27928,      21712,       6720;\n  6:    5040,    46296,    176296,    356344,     403336,     242448,      60480;\n  7:   40320,   420768,   1873328,   4613712,    6790128,    5972944,    2908320,     604800;\n  8:  362880,  4230432,  21488400,  62130016,  111861984,  128447904,   91877264,   37434720,   6652800;\n  9: 3628800, 46658880, 265649184, 879160960, 1864180032, 2626822848, 2460147488, 1476874368, 515744640, 79833600;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+4)*x+(n+3)*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 + 4)*x + (n + 3)*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 + 4)*x + (n + 3)*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-60c7e36cbe887b7e","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-01af55cf26557a61","e-402fd90d0f58c741"],"text":"A137320 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A137320 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}