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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"fd6b97de49fa747fb3a2af4a3c0ea6bf839d9cc5a9642b904b4aff13a418c616","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-872f4101f77d70d2","role":"sequence","name":"one_plus_derivative of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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,     3;\n  2:   5,    16,      12;\n  3:  12,    69,     116,       60;\n  4:  25,   264,     784,      904,      360;\n  5:  46,   903,    4408,     8760,     7728,      2520;\n  6:  77,  2696,   21588,    67976,   101664,     72816,     20160;\n  7: 120,  7049,   93804,   452460,  1040800,   1246224,    752448,    181440;\n  8: 177, 16432,  358880,  2645616,  8996688,  16267968,  16245504,   8485632,   1814400;\n  9: 250, 34827, 1207152, 13766480, 68112672, 177243696, 263276928, 225400320, 103806720, 19958400;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+3+(2*n^2-6*n+6)*x+(n^2-3*n+3)*x^2)*P(n,x) = -((n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2+(4*n^6-36*n^5+128*n^4-236*n^3+252*n^2-160*n+48)*x^3+(6*n^6-54*n^5+192*n^4-354*n^3+378*n^2-240*n+72)*x^4+(4*n^6-36*n^5+128*n^4-236*n^3+252*n^2-160*n+48)*x^5+(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^6)*P(n-4,x) - ((2*n^4-8*n^3+12*n^2-10*n+4)*x+(10*n^4-42*n^3+68*n^2-60*n+24)*x^2+(18*n^4-78*n^3+132*n^2-120*n+48)*x^3+(14*n^4-62*n^3+108*n^2-100*n+40)*x^4+(4*n^4-18*n^3+32*n^2-30*n+12)*x^5)*P(n-3,x) - (n^2-n+1-(2*n^4-8*n^3+6*n^2-6)*x-(6*n^4-24*n^3+23*n^2-3*n-15)*x^2-(6*n^4-24*n^3+24*n^2-2*n-16)*x^3-(2*n^4-8*n^3+8*n^2-6)*x^4)*P(n-2,x) + (2*n^2-4*n+4+(8*n^2-18*n+18)*x+(10*n^2-24*n+24)*x^2+(4*n^2-10*n+10)*x^3)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 5 + 16*x + 12*x^2, P(3,x) = 12 + 69*x + 116*x^2 + 60*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[2, x_] := 5 + 16*x + 12*x^2;\nP[3, x_] := 12 + 69*x + 116*x^2 + 60*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^3 + (6*n^6 - 54*n^5 + 192*n^4 - 354*n^3 + 378*n^2 - 240*n + 72)*x^4 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^5 + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^6)*P[n - 4, x] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (10*n^4 - 42*n^3 + 68*n^2 - 60*n + 24)*x^2 + (18*n^4 - 78*n^3 + 132*n^2 - 120*n + 48)*x^3 + (14*n^4 - 62*n^3 + 108*n^2 - 100*n + 40)*x^4 + (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12)*x^5)*P[n - 3, x] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 6*n^2 - 6)*x - (6*n^4 - 24*n^3 + 23*n^2 - 3*n - 15)*x^2 - (6*n^4 - 24*n^3 + 24*n^2 - 2*n - 16)*x^3 - (2*n^4 - 8*n^3 + 8*n^2 - 6)*x^4)*P[n - 2, x] + (2*n^2 - 4*n + 4 + (8*n^2 - 18*n + 18)*x + (10*n^2 - 24*n + 24)*x^2 + (4*n^2 - 10*n + 10)*x^3)*P[n - 1, x])/(n^2 - 3*n + 3 + (2*n^2 - 6*n + 6)*x + (n^2 - 3*n + 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; v[3] = 5 + 16*x + 12*x^2; v[4] = 12 + 69*x + 116*x^2 + 60*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-((n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^3 + (6*n^6 - 54*n^5 + 192*n^4 - 354*n^3 + 378*n^2 - 240*n + 72)*x^4 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^5 + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^6)*v[i - 4] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (10*n^4 - 42*n^3 + 68*n^2 - 60*n + 24)*x^2 + (18*n^4 - 78*n^3 + 132*n^2 - 120*n + 48)*x^3 + (14*n^4 - 62*n^3 + 108*n^2 - 100*n + 40)*x^4 + (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12)*x^5)*v[i - 3] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 6*n^2 - 6)*x - (6*n^4 - 24*n^3 + 23*n^2 - 3*n - 15)*x^2 - (6*n^4 - 24*n^3 + 24*n^2 - 2*n - 16)*x^3 - (2*n^4 - 8*n^3 + 8*n^2 - 6)*x^4)*v[i - 2] + (2*n^2 - 4*n + 4 + (8*n^2 - 18*n + 18)*x + (10*n^2 - 24*n + 24)*x^2 + (4*n^2 - 10*n + 10)*x^3)*v[i - 1])/(n^2 - 3*n + 3 + (2*n^2 - 6*n + 6)*x + (n^2 - 3*n + 3)*x^2)); 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-08ac2c6d9914754c","steps":["D+1","x \u21a6 x/(x+1)"],"edges":["e-966c24255e4e84d9","e-4b2dbac6e60546ca"],"text":"A154986 \u2192 D+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A154986 \u2192 D+1 \u2192 x \u21a6 x/(x+1)","chain":null}