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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":"5e437d323e700bb821aaa7fd19ba760038ce4efdc1ab9724e5efaa0d5a3e2ac8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-831045384d9c95c9","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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:      2,       5;\n   3:      8,      40,       41;\n   4:     32,     240,      546,       365;\n   5:    128,    1280,     4512,      6560,      3281;\n   6:    512,    6400,    30560,     69200,     73810,      29525;\n   7:   2048,   30720,   185088,    570880,    946632,     797160,     265721;\n   8:   8192,  143360,  1042944,   4076800,   9220288,   12032160,    8370194,    2391485;\n   9:  32768,  655360,  5586944,  26460160,  75983360,  135193600,  145282688,   86093440,  21523361;\n  10: 131072, 2949120, 28827648, 160419840, 559148544, 1263916800, 1850084160, 1688911200, 871696098, 193710245;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = (4*n-8+(23*n-46)*x+(42*n-84)*x^2+(31*n-62)*x^3+(8*n-16)*x^4)*P(n-1,x) - (2*x+14*x^2+30*x^3+26*x^4+8*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 5*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (23*n - 46)*x + (42*n - 84)*x^2 + (31*n - 62)*x^3 + (8*n - 16)*x^4)*P[n - 1, x] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*D[P[n - 1, x], x])/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 5*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (23*n - 46)*x + (42*n - 84)*x^2 + (31*n - 62)*x^3 + (8*n - 16)*x^4)*v[i - 1] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*deriv(v[i - 1], x))/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)); 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-f7935c36a3c0ec8c","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-3b5bae8e6d048f8c","e-6be69665c773c0c7"],"text":"A086645 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A086645 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}