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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"c1ad73d399d6239d6f406174b942195bd89c6d36e2e1f25b923c76360cf803b7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-b8808e9caf8dcf35","role":"sequence","name":"Triangle read by rows, (2, 3)-Lah numbers.","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:                 37,                  1;\n  2:               1865,                102,                 1;\n  3:             122961,              10267,               203,               1;\n  4:           10278281,            1168460,             35102,             348,              1;\n  5:         1062714245,          154447181,           6284362,           94370,            545,            1;\n  6:       133215991777,        23772339810,        1230439231,        24938492,         217695,          802,         1;\n  7:     19913088181377,      4241107108903,      268341129461,      6893548643,       81018467,       450485,      1127,       1;\n  8:   3500271565033393,    871016335435864,    65528274612492,   2050686578056,    30467772758,    227676328,    858732,    1528,    1;\n  9: 714944028588443461, 204425020518981497, 17915166495346948, 664497423961412, 11994858607046, 113087610958, 572466452, 1534452, 2013, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+10*n+26+x)*P(n-1,x) + (2*n+11+(2*n+11)*x)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^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[(n^2 + 10*n + 26 + x)*P[n - 1, x] + (2*n + 11 + (2*n + 11)*x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}]];\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^2 + 10*n + 26 + x)*v[i - 1] + (2*n + 11 + (2*n + 11)*x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 1], x), 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-b8808e9caf8dcf35","steps":["x \u21a6 x+1"],"edges":["e-3333a9756ae82eb9"],"text":"A371259 \u2192 x \u21a6 x+1"},"headline":"A371259 \u2192 x \u21a6 x+1","chain":null}
