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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"7144785e671146c8eee17db3be97f32b344da7114d6f79206f1aec5118f03f3e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6a6e9e081117dc37","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.","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,            2;\n  2:         48,           52,             4;\n  3:        572,         1692,           588,              8;\n  4:       6452,        51556,         41868,           6500,             16;\n  5:      71436,      1347092,       2551376,         923460,          71564,             32;\n  6:     787012,     31727412,     126455088,       98875880,       19404972,         787332,            64;\n  7:    8660140,    701442404,    5328558468,     8303612120,     3392678404,      398351268,       8660908,          128;\n  8:   95268708,  14920837636,  200736794116,   569300953404,   448421586764,   108918044060,    8079663756,     95270500,        256;\n  9: 1047972428, 309944254356, 7006192228872, 33583551401436, 46949455058328, 21528397351692, 3365051684424, 162831916548, 1047976524, 512;\n","recurrence_sentence":"The row polynomials satisfy (9*n-25+18*x)*P(n,x) = -(198*n-352+(324*n^2-1638*n+2284)*x+(729*n^3-5346*n^2+12906*n-11116)*x^2+(2187*n^2-10692*n+12420)*x^3+(1458*n-2592)*x^4)*P(n-2,x) - ((162*n-288)*x+(729*n^2-3483*n+4212)*x^2-(729*n^2-5508*n+9270)*x^3-(2187*n-6804)*x^4-1458*x^5)*d/dx P(n-2,x) + (117*n-307+(162*n^2-819*n+1259)*x+(405*n-1044)*x^2+162*x^3)*P(n-1,x) + ((81*n-225)*x-(81*n-387)*x^2-162*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(198*n - 352 + (324*n^2 - 1638*n + 2284)*x + (729*n^3 - 5346*n^2 + 12906*n - 11116)*x^2 + (2187*n^2 - 10692*n + 12420)*x^3 + (1458*n - 2592)*x^4)*P[n - 2, x] - ((162*n - 288)*x + (729*n^2 - 3483*n + 4212)*x^2 - (729*n^2 - 5508*n + 9270)*x^3 - (2187*n - 6804)*x^4 - 1458*x^5)*D[P[n - 2, x], x] + (117*n - 307 + (162*n^2 - 819*n + 1259)*x + (405*n - 1044)*x^2 + 162*x^3)*P[n - 1, x] + ((81*n - 225)*x - (81*n - 387)*x^2 - 162*x^3)*D[P[n - 1, x], x])/(9*n - 25 + 18*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(198*n - 352 + (324*n^2 - 1638*n + 2284)*x + (729*n^3 - 5346*n^2 + 12906*n - 11116)*x^2 + (2187*n^2 - 10692*n + 12420)*x^3 + (1458*n - 2592)*x^4)*v[i - 2] - ((162*n - 288)*x + (729*n^2 - 3483*n + 4212)*x^2 - (729*n^2 - 5508*n + 9270)*x^3 - (2187*n - 6804)*x^4 - 1458*x^5)*deriv(v[i - 2], x) + (117*n - 307 + (162*n^2 - 819*n + 1259)*x + (405*n - 1044)*x^2 + 162*x^3)*v[i - 1] + ((81*n - 225)*x - (81*n - 387)*x^2 - 162*x^3)*deriv(v[i - 1], x))/(9*n - 25 + 18*x)); 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-6a6e9e081117dc37","steps":["D+1"],"edges":["e-db4efe3d6daa88cf"],"text":"A257619 \u2192 D+1"},"headline":"A257619 \u2192 D+1","chain":null}