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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"5ada066789cc98cb58db8d0984c26206398d877b718ed4b0bab510bd9eb5a515","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c16b107c87dd2b64","role":"sequence","name":"basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:     4,       7;\n  2:    13,      50,      46;\n  3:    43,     255,     489,      304;\n  4:   142,    1138,    3345,     4276,      2008;\n  5:   469,    4736,   18799,    36667,     35156,     13264;\n  6:  1549,   18870,   94416,   248348,    362238,    277872,     87616;\n  7:  5116,   72985,  440823,  1461146,   2870489,   3342756,   2137024,    578752;\n  8: 16897,  276266, 1955341,  7824284,  19359985,  30333656,  29393536,  16107968,   3822976;\n  9: 55807, 1028745, 8349831, 39162099, 116965170, 230700663, 300515304, 249320016, 119558592, 25252864;\n","recurrence_sentence":"The row polynomials satisfy (9*n+9+(27*n+27)*x+(27*n+27)*x^2+(9*n+9)*x^3)*P(n,x) = (79*n^2-300*n+293+(540*n^2-2133*n+1935)*x+(1578*n^2-6183*n+5199)*x^2+(2492*n^2-9375*n+7333)*x^3+(2223*n^2-7857*n+5760)*x^4+(1056*n^2-3456*n+2400)*x^5+(208*n^2-624*n+416)*x^6)*P(n-1,x) - (65*n-175+(644*n-1720)*x+(2634*n-6834)*x^2+(5812*n-14468)*x^3+(7513*n-17807)*x^4+(5712*n-12828)*x^5+(2372*n-5032)*x^6+(416*n-832)*x^7)*d/dx P(n-1,x) + (13+173*x+946*x^2+2798*x^3+4945*x^4+5393*x^5+3568*x^6+1316*x^7+208*x^8)*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[Together[((79*n^2 - 300*n + 293 + (540*n^2 - 2133*n + 1935)*x + (1578*n^2 - 6183*n + 5199)*x^2 + (2492*n^2 - 9375*n + 7333)*x^3 + (2223*n^2 - 7857*n + 5760)*x^4 + (1056*n^2 - 3456*n + 2400)*x^5 + (208*n^2 - 624*n + 416)*x^6)*P[n - 1, x] - (65*n - 175 + (644*n - 1720)*x + (2634*n - 6834)*x^2 + (5812*n - 14468)*x^3 + (7513*n - 17807)*x^4 + (5712*n - 12828)*x^5 + (2372*n - 5032)*x^6 + (416*n - 832)*x^7)*D[P[n - 1, x], x] + (13 + 173*x + 946*x^2 + 2798*x^3 + 4945*x^4 + 5393*x^5 + 3568*x^6 + 1316*x^7 + 208*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 9 + (27*n + 27)*x + (27*n + 27)*x^2 + (9*n + 9)*x^3)]];\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] = ((79*n^2 - 300*n + 293 + (540*n^2 - 2133*n + 1935)*x + (1578*n^2 - 6183*n + 5199)*x^2 + (2492*n^2 - 9375*n + 7333)*x^3 + (2223*n^2 - 7857*n + 5760)*x^4 + (1056*n^2 - 3456*n + 2400)*x^5 + (208*n^2 - 624*n + 416)*x^6)*v[i - 1] - (65*n - 175 + (644*n - 1720)*x + (2634*n - 6834)*x^2 + (5812*n - 14468)*x^3 + (7513*n - 17807)*x^4 + (5712*n - 12828)*x^5 + (2372*n - 5032)*x^6 + (416*n - 832)*x^7)*deriv(v[i - 1], x) + (13 + 173*x + 946*x^2 + 2798*x^3 + 4945*x^4 + 5393*x^5 + 3568*x^6 + 1316*x^7 + 208*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 9 + (27*n + 27)*x + (27*n + 27)*x^2 + (9*n + 9)*x^3)); 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-d6b7687ba0ff59dd","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-dddeff0317c778fa","e-85bf1b24f11fb972"],"text":"A116412 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A116412 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}