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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":"4abfa1c94de7f92c7b6905319c831f61af469d60fceee21ca0f6b009a3de147d","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-20086f90792ae22f","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:      5,       9;\n  2:     17,      58,       49;\n  3:     61,     307,      511,      281;\n  4:    217,    1444,     3582,     3924,      1601;\n  5:    773,    6397,    21074,    34538,     28153,      9129;\n  6:   2753,   27246,   111903,   244100,    298223,    193454,     52049;\n  7:   9805,  112935,   555561,  1512923,   2462999,   2396805,   1290803,    296761;\n  8:  34921,  458840,  2629652,  8585176,  17462270,  22658056,  18314148,   8430408,  1692001;\n  9: 124373, 1835825, 12011108, 45714740, 111538470, 180908814, 195047156, 134787524, 54171821, 9647049;\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) = (38*n^2-100*n+62+(260*n^2-722*n+462)*x+(742*n^2-2138*n+1396)*x^2+(1128*n^2-3328*n+2200)*x^3+(962*n^2-2876*n+1914)*x^4+(436*n^2-1310*n+874)*x^5+(82*n^2-246*n+164)*x^6)*P(n-1,x) - (51*n-98+(434*n-846)*x+(1565*n-3084)*x^2+(3104*n-6164)*x^3+(3661*n-7306)*x^4+(2570*n-5142)*x^5+(995*n-1992)*x^6+(164*n-328)*x^7)*d/dx P(n-1,x) + (17+173*x+759*x^2+1877*x^3+2865*x^4+2767*x^5+1653*x^6+559*x^7+82*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 9*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 9*x;\nP[n_, x_] := P[n, x] = Expand[Together[((38*n^2 - 100*n + 62 + (260*n^2 - 722*n + 462)*x + (742*n^2 - 2138*n + 1396)*x^2 + (1128*n^2 - 3328*n + 2200)*x^3 + (962*n^2 - 2876*n + 1914)*x^4 + (436*n^2 - 1310*n + 874)*x^5 + (82*n^2 - 246*n + 164)*x^6)*P[n - 1, x] - (51*n - 98 + (434*n - 846)*x + (1565*n - 3084)*x^2 + (3104*n - 6164)*x^3 + (3661*n - 7306)*x^4 + (2570*n - 5142)*x^5 + (995*n - 1992)*x^6 + (164*n - 328)*x^7)*D[P[n - 1, x], x] + (17 + 173*x + 759*x^2 + 1877*x^3 + 2865*x^4 + 2767*x^5 + 1653*x^6 + 559*x^7 + 82*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] = 5 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = ((38*n^2 - 100*n + 62 + (260*n^2 - 722*n + 462)*x + (742*n^2 - 2138*n + 1396)*x^2 + (1128*n^2 - 3328*n + 2200)*x^3 + (962*n^2 - 2876*n + 1914)*x^4 + (436*n^2 - 1310*n + 874)*x^5 + (82*n^2 - 246*n + 164)*x^6)*v[i - 1] - (51*n - 98 + (434*n - 846)*x + (1565*n - 3084)*x^2 + (3104*n - 6164)*x^3 + (3661*n - 7306)*x^4 + (2570*n - 5142)*x^5 + (995*n - 1992)*x^6 + (164*n - 328)*x^7)*deriv(v[i - 1], x) + (17 + 173*x + 759*x^2 + 1877*x^3 + 2865*x^4 + 2767*x^5 + 1653*x^6 + 559*x^7 + 82*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-e646c30d067149e7","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-4395b7c136181822","e-4b586df1aae103ad"],"text":"A249252 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A249252 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}