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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":"03843fd4a2bd1d50122c7b389f06b5f57aa13abcae4a3f9ca7660179dc4dfa9b","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-59c36108e7f99c2f","role":"sequence","name":"basis_map_A013610 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:     3,      6;\n  2:    10,     41,      40;\n  3:    33,    204,     408,      264;\n  4:   109,    901,    2733,     3604,     1744;\n  5:   360,   3726,   15165,    30330,    29808,     11520;\n  6:  1189,  14784,   75525,   202853,   302100,    236544,     76096;\n  7:  3927,  57012,  350514,  1182768,  2365536,   2804112,   1824384,    502656;\n  8: 12970, 215327, 1547815,  6291002, 15811681,  25164008,  24765040,  13780928,   3320320;\n  9: 42837, 800448, 6586689, 31323282, 94862487, 189724974, 250586256, 210774048, 102457344, 21932544;\n","recurrence_sentence":"The row polynomials satisfy (9*n+27*n*x+27*n*x^2+9*n*x^3)*P(n,x) = (79*n^2-216*n+164+(540*n^2-1641*n+1236)*x+(1578*n^2-5055*n+3720)*x^2+(2492*n^2-8103*n+5800)*x^3+(2223*n^2-7149*n+4980)*x^4+(1056*n^2-3300*n+2244)*x^5+(208*n^2-624*n+416)*x^6)*P(n-1,x) - (65*n-136+(644*n-1396)*x+(2634*n-5772)*x^2+(5812*n-12680)*x^3+(7513*n-16160)*x^4+(5712*n-12036)*x^5+(2372*n-4876)*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 - 216*n + 164 + (540*n^2 - 1641*n + 1236)*x + (1578*n^2 - 5055*n + 3720)*x^2 + (2492*n^2 - 8103*n + 5800)*x^3 + (2223*n^2 - 7149*n + 4980)*x^4 + (1056*n^2 - 3300*n + 2244)*x^5 + (208*n^2 - 624*n + 416)*x^6)*P[n - 1, x] - (65*n - 136 + (644*n - 1396)*x + (2634*n - 5772)*x^2 + (5812*n - 12680)*x^3 + (7513*n - 16160)*x^4 + (5712*n - 12036)*x^5 + (2372*n - 4876)*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 + 27*n*x + 27*n*x^2 + 9*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] = ((79*n^2 - 216*n + 164 + (540*n^2 - 1641*n + 1236)*x + (1578*n^2 - 5055*n + 3720)*x^2 + (2492*n^2 - 8103*n + 5800)*x^3 + (2223*n^2 - 7149*n + 4980)*x^4 + (1056*n^2 - 3300*n + 2244)*x^5 + (208*n^2 - 624*n + 416)*x^6)*v[i - 1] - (65*n - 136 + (644*n - 1396)*x + (2634*n - 5772)*x^2 + (5812*n - 12680)*x^3 + (7513*n - 16160)*x^4 + (5712*n - 12036)*x^5 + (2372*n - 4876)*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 + 27*n*x + 27*n*x^2 + 9*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-d5c14b9fec875ecc","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-16250729b41a60db","e-a00d483f8887d7f2"],"text":"A121574 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A121574 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}