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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":"dd034c4ec733d9df1c923303f7871e80ea3f3a81d623545b616c75ff56dd0816","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-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 of Pascal-(1,5,1) array.","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:     2,       6;\n  2:     9,      54,       61;\n  3:    28,     252,      676,       516;\n  4:   101,    1212,     5014,      8268,      4621;\n  5:   342,    5130,    28940,     75780,     91230,      40626;\n  6:  1189,   21402,   152615,    547260,   1033635,     973962,     359281;\n  7:  4088,   85848,   740792,   3385560,   8803880,   12984264,   10061864,    3171336;\n  8: 14121,  338904,  3431532,  19065672,  63322470,  128352168,  154885452,  101841336,   28010041;\n  9: 48682, 1314414, 15278088, 100018296, 405144972, 1050315876, 1739818152, 1775314584, 1013754778, 247343646;\n","recurrence_sentence":"The row polynomials satisfy (2*n+6*n*x+6*n*x^2+2*n*x^3)*P(n,x) = (27*n^2-85*n+62+(198*n^2-654*n+480)*x+(607*n^2-2029*n+1470)*x^2+(988*n^2-3268*n+2320)*x^3+(897*n^2-2895*n+2010)*x^4+(430*n^2-1342*n+912)*x^5+(85*n^2-255*n+170)*x^6)*P(n-1,x) - (18*n-42+(210*n-486)*x+(942*n-2140)*x^2+(2198*n-4880)*x^3+(2942*n-6370)*x^4+(2286*n-4822)*x^5+(962*n-1976)*x^6+(170*n-340)*x^7)*d/dx P(n-1,x) + (3+48*x+302*x^2+980*x^3+1840*x^4+2088*x^5+1418*x^6+532*x^7+85*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[((27*n^2 - 85*n + 62 + (198*n^2 - 654*n + 480)*x + (607*n^2 - 2029*n + 1470)*x^2 + (988*n^2 - 3268*n + 2320)*x^3 + (897*n^2 - 2895*n + 2010)*x^4 + (430*n^2 - 1342*n + 912)*x^5 + (85*n^2 - 255*n + 170)*x^6)*P[n - 1, x] - (18*n - 42 + (210*n - 486)*x + (942*n - 2140)*x^2 + (2198*n - 4880)*x^3 + (2942*n - 6370)*x^4 + (2286*n - 4822)*x^5 + (962*n - 1976)*x^6 + (170*n - 340)*x^7)*D[P[n - 1, x], x] + (3 + 48*x + 302*x^2 + 980*x^3 + 1840*x^4 + 2088*x^5 + 1418*x^6 + 532*x^7 + 85*x^8)*D[P[n - 1, x], {x, 2}])/(2*n + 6*n*x + 6*n*x^2 + 2*n*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] = ((27*n^2 - 85*n + 62 + (198*n^2 - 654*n + 480)*x + (607*n^2 - 2029*n + 1470)*x^2 + (988*n^2 - 3268*n + 2320)*x^3 + (897*n^2 - 2895*n + 2010)*x^4 + (430*n^2 - 1342*n + 912)*x^5 + (85*n^2 - 255*n + 170)*x^6)*v[i - 1] - (18*n - 42 + (210*n - 486)*x + (942*n - 2140)*x^2 + (2198*n - 4880)*x^3 + (2942*n - 6370)*x^4 + (2286*n - 4822)*x^5 + (962*n - 1976)*x^6 + (170*n - 340)*x^7)*deriv(v[i - 1], x) + (3 + 48*x + 302*x^2 + 980*x^3 + 1840*x^4 + 2088*x^5 + 1418*x^6 + 532*x^7 + 85*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n + 6*n*x + 6*n*x^2 + 2*n*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-6ad01dc6e705cdb6","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-09b3cf169dcfe004","e-c2c394e25de6742c"],"text":"A081580 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A081580 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}