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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":"946b8da1a961f100c1548b8642b8aeed1c37da957ad2763492ad51d449cb009f","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-bfe76bb735cd71bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,3,1) 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,      3;\n  2:     7,     21,      15;\n  3:    20,     90,     132,      63;\n  4:    61,    366,     810,     783,     279;\n  5:   182,   1365,    4044,    5913,    4266,     1215;\n  6:   547,   4923,   18273,   35793,   39015,    22437,     5319;\n  7:  1640,  17220,   76824,  188730,  275688,   239436,   114480,   23247;\n  8:  4921,  59052,  307716,  909306, 1666386,  1939140,  1399248,  572427,  101655;\n  9: 14762, 199287, 1187880, 4102686, 9047268, 13209318, 12768408, 7879275, 2816694, 444447;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (36*n^2-115*n+81+(240*n^2-768*n+537)*x+(666*n^2-2121*n+1470)*x^2+(984*n^2-3103*n+2130)*x^3+(816*n^2-2538*n+1725)*x^4+(360*n^2-1101*n+741)*x^5+(66*n^2-198*n+132)*x^6)*P(n-1,x) - (48*n-104+(400*n-859)*x+(1411*n-2999)*x^2+(2736*n-5750)*x^3+(3154*n-6550)*x^4+(2164*n-4439)*x^5+(819*n-1659)*x^6+(132*n-264)*x^7)*d/dx P(n-1,x) + (16+160*x+689*x^2+1671*x^3+2500*x^4+2366*x^5+1385*x^6+459*x^7+66*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[((36*n^2 - 115*n + 81 + (240*n^2 - 768*n + 537)*x + (666*n^2 - 2121*n + 1470)*x^2 + (984*n^2 - 3103*n + 2130)*x^3 + (816*n^2 - 2538*n + 1725)*x^4 + (360*n^2 - 1101*n + 741)*x^5 + (66*n^2 - 198*n + 132)*x^6)*P[n - 1, x] - (48*n - 104 + (400*n - 859)*x + (1411*n - 2999)*x^2 + (2736*n - 5750)*x^3 + (3154*n - 6550)*x^4 + (2164*n - 4439)*x^5 + (819*n - 1659)*x^6 + (132*n - 264)*x^7)*D[P[n - 1, x], x] + (16 + 160*x + 689*x^2 + 1671*x^3 + 2500*x^4 + 2366*x^5 + 1385*x^6 + 459*x^7 + 66*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^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] = ((36*n^2 - 115*n + 81 + (240*n^2 - 768*n + 537)*x + (666*n^2 - 2121*n + 1470)*x^2 + (984*n^2 - 3103*n + 2130)*x^3 + (816*n^2 - 2538*n + 1725)*x^4 + (360*n^2 - 1101*n + 741)*x^5 + (66*n^2 - 198*n + 132)*x^6)*v[i - 1] - (48*n - 104 + (400*n - 859)*x + (1411*n - 2999)*x^2 + (2736*n - 5750)*x^3 + (3154*n - 6550)*x^4 + (2164*n - 4439)*x^5 + (819*n - 1659)*x^6 + (132*n - 264)*x^7)*deriv(v[i - 1], x) + (16 + 160*x + 689*x^2 + 1671*x^3 + 2500*x^4 + 2366*x^5 + 1385*x^6 + 459*x^7 + 66*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^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-d8f9ad9ce0e039d0","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-118ee1d5972a9fb3","e-01e72d91a125f706"],"text":"A081578 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A081578 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}