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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":"23c1dde4dc4c9b58804b278b5a1cdc4b9f37bcb122f168658a15a41a4ff7ed77","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-bf2f08bd0370af8d","role":"sequence","name":"basis_map_A013609 of Pascal-(1,4,1) 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,      4;\n  2:     8,     32,      28;\n  3:    24,    144,     272,      160;\n  4:    80,    640,    1840,     2240,      976;\n  5:   256,   2560,    9920,    18560,    16736,     5824;\n  6:   832,   9984,   48640,   122880,   169536,   121088,     35008;\n  7:  2688,  37632,  220928,   704000,  1313536,  1434112,    848384,   209920;\n  8:  8704, 139264,  956672,  3681280,  8670976, 12793856,  11545088,  5826560,  1259776;\n  9: 28160, 506880, 3988480, 17991680, 51236352, 95472640, 116367360, 89456640, 39365120, 7558144;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (20*n^2-64*n+46+(140*n^2-456*n+326)*x+(408*n^2-1330*n+940)*x^2+(632*n^2-2038*n+1420)*x^3+(548*n^2-1734*n+1190)*x^4+(252*n^2-778*n+526)*x^5+(48*n^2-144*n+96)*x^6)*P(n-1,x) - (20*n-45+(190*n-423)*x+(744*n-1632)*x^2+(1572*n-3390)*x^3+(1948*n-4125)*x^4+(1422*n-2955)*x^5+(568*n-1158)*x^6+(96*n-192)*x^7)*d/dx P(n-1,x) + (5+60*x+301*x^2+828*x^3+1375*x^4+1420*x^5+895*x^6+316*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[((20*n^2 - 64*n + 46 + (140*n^2 - 456*n + 326)*x + (408*n^2 - 1330*n + 940)*x^2 + (632*n^2 - 2038*n + 1420)*x^3 + (548*n^2 - 1734*n + 1190)*x^4 + (252*n^2 - 778*n + 526)*x^5 + (48*n^2 - 144*n + 96)*x^6)*P[n - 1, x] - (20*n - 45 + (190*n - 423)*x + (744*n - 1632)*x^2 + (1572*n - 3390)*x^3 + (1948*n - 4125)*x^4 + (1422*n - 2955)*x^5 + (568*n - 1158)*x^6 + (96*n - 192)*x^7)*D[P[n - 1, x], x] + (5 + 60*x + 301*x^2 + 828*x^3 + 1375*x^4 + 1420*x^5 + 895*x^6 + 316*x^7 + 48*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] = ((20*n^2 - 64*n + 46 + (140*n^2 - 456*n + 326)*x + (408*n^2 - 1330*n + 940)*x^2 + (632*n^2 - 2038*n + 1420)*x^3 + (548*n^2 - 1734*n + 1190)*x^4 + (252*n^2 - 778*n + 526)*x^5 + (48*n^2 - 144*n + 96)*x^6)*v[i - 1] - (20*n - 45 + (190*n - 423)*x + (744*n - 1632)*x^2 + (1572*n - 3390)*x^3 + (1948*n - 4125)*x^4 + (1422*n - 2955)*x^5 + (568*n - 1158)*x^6 + (96*n - 192)*x^7)*deriv(v[i - 1], x) + (5 + 60*x + 301*x^2 + 828*x^3 + 1375*x^4 + 1420*x^5 + 895*x^6 + 316*x^7 + 48*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-ed9434bfd889c409","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-175e767cc5e2f6ad","e-9bb12cfb8eacc33f"],"text":"A081579 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A081579 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}