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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":"92189e80661e3aeb563876f81807b3235adb5a66db10a0431dcd68ac2b9d86b3","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-1a0f876fa1441737","role":"sequence","name":"basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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":[{"id":"e-9c5e956579680e26","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-706703e52ab62d2a","conclusion_name":"Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3."}],"used_by_total":1,"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:        -1,         1;\n  2:        -2,         1,        1;\n  3:       -10,         3,        6,        1;\n  4:       -80,        14,       51,       14,       1;\n  5:      -880,        74,      575,      205,      25,       1;\n  6:    -12320,       156,     8124,     3445,     555,      39,     1;\n  7:   -209440,     -9668,   138264,    66689,   12880,    1218,    56,    1;\n  8:  -4188800,   -402800,  2755612,  1472044,  324289,   37240,  2338,   76,  1;\n  9: -96342400, -13453200, 62976276, 36612624, 8930691, 1180809, 91014, 4086, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-4+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n - 4 + x)*P[n - 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] = (3*n - 4 + x)*v[i - 1]); 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-1a0f876fa1441737","steps":["x \u21a6 x\u22121"],"edges":["e-c5f7781ad297615f"],"text":"basis-map coefficients of A371080, shifted \u2192 x \u21a6 x\u22121"},"headline":"basis-map coefficients of A371080, shifted \u2192 x \u21a6 x\u22121","chain":null}