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Triangle read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,        2;\n   3:      2,       10,         9;\n   4:      6,       56,       109,         60;\n   5:     24,      368,      1148,       1316,        513;\n   6:    120,     2800,     12340,      22000,      17625,       5286;\n   7:    720,    24240,    140680,     345760,     426190,     261006,      63417;\n   8:   5040,   235200,   1718920,    5449920,    9175530,    8596952,    4248405,     865824;\n   9:  40320,  2526720,  22547840,   88415040,  190426320,  242437216,  182623896,   75488904,   13229505;\n  10: 362880, 29756160, 317036160, 1493313920, 3946751760, 6396530784, 6524724808, 4100457888, 1454519601, 223336458;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n-1-(n-5)*x-(5*n-12)*x^2-(3*n-6)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 - (n - 5)*x - (5*n - 12)*x^2 - (3*n - 6)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n - 1 - (n - 5)*x - (5*n - 12)*x^2 - (3*n - 6)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e9164c616dcc1bbd","steps":["x \u21a6 x/(x+1)"],"edges":["e-f379195edda1b718"],"text":"A322944 \u2192 x \u21a6 x/(x+1)"},"headline":"A322944 \u2192 x \u21a6 x/(x+1)","chain":null}