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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":"8b0947d552d466d5d99a2dbab78fd3358ba9050238c6d7e95ed255f2b99ce703","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"real_rooted","from_index":2,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[{"id":"e-f961be0564a72cf6","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-88fcd1b546ee4111","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n)."},{"id":"e-f961be0564a72cf6","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-88fcd1b546ee4111","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n)."}],"used_by_total":2,"links":[],"oeis":[{"oeis_id":"A051142","relation":"tail"}],"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:               32,             -12,               1;\n  1:             -384,             176,             -24,             1;\n  2:             6144,           -3200,             560,           -40,             1;\n  3:          -122880,           70144,          -14400,          1360,           -60,           1;\n  4:          2949120,        -1806336,          415744,        -47040,          2800,         -84,           1;\n  5:        -82575360,        53526528,       -13447168,       1732864,       -125440,        5152,        -112,        1;\n  6:       2642411520,     -1795424256,       483835904,     -68898816,       5746944,     -290304,        8736,     -144,        1;\n  7:     -95126814720,     67277684736,    -19213516800,    2964193280,    -275788800,    16197888,     -604800,    13920,     -180,     1;\n  8:    3805072588800,  -2786234204160,    835818356736, -137781248000,   13995745280,  -923704320,    40389888, -1161600,    21120,  -220,    1;\n  9: -167423193907200, 126399377571840, -39562241900544, 6898193268736, -753594040320, 54638735360, -2700859392, 91500288, -2090880, 30800, -264, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n+8-x)*P(n-1,x) for n >= 1, with P(0,x) = 32 - 12*x + x^2.","mathematica":"P[0, x_] := 32 - 12*x + x^2;\nP[n_, x_] := P[n, x] = Expand[-(4*n + 8 - 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] = 32 - 12*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(4*n + 8 - 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-4f6e747b794bcc2e","steps":["x \u21a6 (\u22121/2)x+1 (shift +2)"],"edges":["e-41f01d4be10aeae3"],"text":"A028338 \u2192 x \u21a6 (\u22121/2)x+1 (shift +2)"},"headline":"A028338 \u2192 x \u21a6 (\u22121/2)x+1 (shift +2)","chain":null}