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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":"8ab7246e8ea63b848e543a7529a4fd9e40fd60f4013c529409c2c84ecad158aa","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-131f5baf2477adf2","role":"sequence","name":"basis_map_A007318 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,      5;\n  2:    13,     34,      22;\n  3:    43,    171,     225,       98;\n  4:   142,    758,    1509,     1328,      436;\n  5:   469,   3142,    8383,    11135,     7364,     1940;\n  6:  1549,  12486,   41784,    74308,    74070,    39240,     8632;\n  7:  5116,  48203,  194043,   432628,   576977,   460302,   203404,    38408;\n  8: 16897, 182206,  857281,  2298688,  3842005,  4098880,  2725912,  1033216,  170896;\n  9: 55807, 677757, 3649551, 11436351, 22983714, 30721149, 27311520, 15572520, 5167584, 760400;\n","recurrence_sentence":"The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (29*n^2-104*n+83+(188*n^2-667*n+513)*x+(510*n^2-1773*n+1317)*x^2+(740*n^2-2501*n+1799)*x^3+(605*n^2-1975*n+1380)*x^4+(264*n^2-828*n+564)*x^5+(48*n^2-144*n+96)*x^6)*P(n-1,x) - (39*n-93+(316*n-740)*x+(1090*n-2498)*x^2+(2076*n-4644)*x^3+(2359*n-5141)*x^4+(1600*n-3392)*x^5+(600*n-1236)*x^6+(96*n-192)*x^7)*d/dx P(n-1,x) + (13+127*x+536*x^2+1278*x^3+1885*x^4+1763*x^5+1022*x^6+336*x^7+48*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[((29*n^2 - 104*n + 83 + (188*n^2 - 667*n + 513)*x + (510*n^2 - 1773*n + 1317)*x^2 + (740*n^2 - 2501*n + 1799)*x^3 + (605*n^2 - 1975*n + 1380)*x^4 + (264*n^2 - 828*n + 564)*x^5 + (48*n^2 - 144*n + 96)*x^6)*P[n - 1, x] - (39*n - 93 + (316*n - 740)*x + (1090*n - 2498)*x^2 + (2076*n - 4644)*x^3 + (2359*n - 5141)*x^4 + (1600*n - 3392)*x^5 + (600*n - 1236)*x^6 + (96*n - 192)*x^7)*D[P[n - 1, x], x] + (13 + 127*x + 536*x^2 + 1278*x^3 + 1885*x^4 + 1763*x^5 + 1022*x^6 + 336*x^7 + 48*x^8)*D[P[n - 1, x], {x, 2}])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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] = ((29*n^2 - 104*n + 83 + (188*n^2 - 667*n + 513)*x + (510*n^2 - 1773*n + 1317)*x^2 + (740*n^2 - 2501*n + 1799)*x^3 + (605*n^2 - 1975*n + 1380)*x^4 + (264*n^2 - 828*n + 564)*x^5 + (48*n^2 - 144*n + 96)*x^6)*v[i - 1] - (39*n - 93 + (316*n - 740)*x + (1090*n - 2498)*x^2 + (2076*n - 4644)*x^3 + (2359*n - 5141)*x^4 + (1600*n - 3392)*x^5 + (600*n - 1236)*x^6 + (96*n - 192)*x^7)*deriv(v[i - 1], x) + (13 + 127*x + 536*x^2 + 1278*x^3 + 1885*x^4 + 1763*x^5 + 1022*x^6 + 336*x^7 + 48*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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 A007318","x \u21a6 x/(x+1)"],"edges":["e-32d1efa81dadfdb2","e-afefd5dda6a5b998"],"text":"A116412 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A116412 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}