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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":"df33777e773d81c00291d52378c2c7eb6c883ba868222f01e713de22f3e9e5a5","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-359eaa7b22014e19","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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,      7;\n  2:    13,     44,      37;\n  3:    43,    216,     360,      199;\n  4:   142,    946,    2355,     2596,     1069;\n  5:   469,   3893,   12889,    21274,    17504,     5743;\n  6:  1549,  15396,   63609,   139820,   172449,   113148,    30853;\n  7:  5116,  59233,  293313,   805232,  1323551,  1302492,   710536,   165751;\n  8: 16897, 223322, 1288999,  4243700,  8715940, 11435366,  9359263,  4368788,   890461;\n  9: 55807, 829032, 5464851, 20979732, 51692193, 84769407, 92519733, 64805022, 26433648, 4783807;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (29*n^2-77*n+48+(200*n^2-562*n+362)*x+(576*n^2-1680*n+1104)*x^2+(884*n^2-2636*n+1752)*x^3+(761*n^2-2293*n+1532)*x^4+(348*n^2-1050*n+702)*x^5+(66*n^2-198*n+132)*x^6)*P(n-1,x) - (39*n-76+(334*n-660)*x+(1213*n-2420)*x^2+(2424*n-4864)*x^3+(2881*n-5796)*x^4+(2038*n-4100)*x^5+(795*n-1596)*x^6+(132*n-264)*x^7)*d/dx P(n-1,x) + (13+133*x+587*x^2+1461*x^3+2245*x^4+2183*x^5+1313*x^6+447*x^7+66*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 7*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 7*x;\nP[n_, x_] := P[n, x] = Expand[Together[((29*n^2 - 77*n + 48 + (200*n^2 - 562*n + 362)*x + (576*n^2 - 1680*n + 1104)*x^2 + (884*n^2 - 2636*n + 1752)*x^3 + (761*n^2 - 2293*n + 1532)*x^4 + (348*n^2 - 1050*n + 702)*x^5 + (66*n^2 - 198*n + 132)*x^6)*P[n - 1, x] - (39*n - 76 + (334*n - 660)*x + (1213*n - 2420)*x^2 + (2424*n - 4864)*x^3 + (2881*n - 5796)*x^4 + (2038*n - 4100)*x^5 + (795*n - 1596)*x^6 + (132*n - 264)*x^7)*D[P[n - 1, x], x] + (13 + 133*x + 587*x^2 + 1461*x^3 + 2245*x^4 + 2183*x^5 + 1313*x^6 + 447*x^7 + 66*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 7*x; for(n = 2, N, my(i = n + 1); v[i] = ((29*n^2 - 77*n + 48 + (200*n^2 - 562*n + 362)*x + (576*n^2 - 1680*n + 1104)*x^2 + (884*n^2 - 2636*n + 1752)*x^3 + (761*n^2 - 2293*n + 1532)*x^4 + (348*n^2 - 1050*n + 702)*x^5 + (66*n^2 - 198*n + 132)*x^6)*v[i - 1] - (39*n - 76 + (334*n - 660)*x + (1213*n - 2420)*x^2 + (2424*n - 4864)*x^3 + (2881*n - 5796)*x^4 + (2038*n - 4100)*x^5 + (795*n - 1596)*x^6 + (132*n - 264)*x^7)*deriv(v[i - 1], x) + (13 + 133*x + 587*x^2 + 1461*x^3 + 2245*x^4 + 2183*x^5 + 1313*x^6 + 447*x^7 + 66*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*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-e7331b0d50084c15","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-212843e2c5c5a487","e-35dcb6fff08d7919"],"text":"A249251 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A249251 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}
