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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":"53d3e85bccf14f9cbb92a37d68755d1b439bb15025a3e0f0aa5f002efa5bd429","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-e6d483569f3966cd","role":"sequence","name":"basis_map_A013610 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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,       10;\n  2:     18,       90,       108;\n  3:     80,      600,      1464,      1160;\n  4:    356,     3560,     13116,     21080,      12464;\n  5:   1584,    19800,     97632,    237240,     283968,     133920;\n  6:   7048,   105720,    653208,   2127080,    3848928,    3668640,     1438912;\n  7:  31360,   548800,   4075968,  16649600,   40387904,   58169280,    46052992,    15460480;\n  8: 139536,  2790720,  24211872, 118990080,  362241936,  699399360,   836260416,   566110080,  166116096;\n  9: 620864, 13969440, 138645504, 796528320, 2918748096, 7073431200, 11335937280, 11583849600, 6848593920, 1784844800;\n","recurrence_sentence":"The row polynomials satisfy (3*n+9*n*x+9*n*x^2+3*n*x^3)*P(n,x) = (25*n^2-75*n+62+(180*n^2-600*n+486)*x+(558*n^2-1932*n+1500)*x^2+(932*n^2-3210*n+2380)*x^3+(873*n^2-2913*n+2070)*x^4+(432*n^2-1374*n+942)*x^5+(88*n^2-264*n+176)*x^6)*P(n-1,x) - (20*n-46+(212*n-496)*x+(918*n-2142)*x^2+(2128*n-4880)*x^3+(2872*n-6410)*x^4+(2268*n-4896)*x^5+(974*n-2026)*x^6+(176*n-352)*x^7)*d/dx P(n-1,x) + (4+56*x+322*x^2+998*x^3+1840*x^4+2084*x^5+1426*x^6+542*x^7+88*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[((25*n^2 - 75*n + 62 + (180*n^2 - 600*n + 486)*x + (558*n^2 - 1932*n + 1500)*x^2 + (932*n^2 - 3210*n + 2380)*x^3 + (873*n^2 - 2913*n + 2070)*x^4 + (432*n^2 - 1374*n + 942)*x^5 + (88*n^2 - 264*n + 176)*x^6)*P[n - 1, x] - (20*n - 46 + (212*n - 496)*x + (918*n - 2142)*x^2 + (2128*n - 4880)*x^3 + (2872*n - 6410)*x^4 + (2268*n - 4896)*x^5 + (974*n - 2026)*x^6 + (176*n - 352)*x^7)*D[P[n - 1, x], x] + (4 + 56*x + 322*x^2 + 998*x^3 + 1840*x^4 + 2084*x^5 + 1426*x^6 + 542*x^7 + 88*x^8)*D[P[n - 1, x], {x, 2}])/(3*n + 9*n*x + 9*n*x^2 + 3*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] = ((25*n^2 - 75*n + 62 + (180*n^2 - 600*n + 486)*x + (558*n^2 - 1932*n + 1500)*x^2 + (932*n^2 - 3210*n + 2380)*x^3 + (873*n^2 - 2913*n + 2070)*x^4 + (432*n^2 - 1374*n + 942)*x^5 + (88*n^2 - 264*n + 176)*x^6)*v[i - 1] - (20*n - 46 + (212*n - 496)*x + (918*n - 2142)*x^2 + (2128*n - 4880)*x^3 + (2872*n - 6410)*x^4 + (2268*n - 4896)*x^5 + (974*n - 2026)*x^6 + (176*n - 352)*x^7)*deriv(v[i - 1], x) + (4 + 56*x + 322*x^2 + 998*x^3 + 1840*x^4 + 2084*x^5 + 1426*x^6 + 542*x^7 + 88*x^8)*deriv(deriv(v[i - 1], x), x))/(3*n + 9*n*x + 9*n*x^2 + 3*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-1bffaa6198344f93","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-04dfb33ef5bb2a04","e-04e1988976d6dfd3"],"text":"A059473 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A059473 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}