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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":"a854471ae070876ab84bcde0f396530f4fc82dbc64ba219fdd9f9baab24a4148","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:          0;\n  1:          2;\n  2:         44,            8;\n  3:        564,         1128,            24;\n  4:       6436,        45120,         19308,             64;\n  5:      71404,      1275688,       1913532,         285616,            160;\n  6:     786948,     30940464,     110984856,       61880928,        3934740,            384;\n  7:    8660012,    692782392,    4982167272,     6642889696,     1731955980,       51960072,           896;\n  8:   95268452,  14825569184,  193324009524,   504859616896,   322206682540,    44476707552,     666879164,       2048;\n  9: 1047971916, 308896282440, 6851744087652, 31299636705552, 39124545881940, 13703488175304, 1081136988540, 8383775328, 4608;\n","recurrence_sentence":"The row polynomials satisfy (9*n-16)*P(n,x) = -(198*n-154+(324*n^2-1152*n+700)*x+(729*n^3-3888*n^2+6183*n-2800)*x^2)*P(n-2,x) - ((162*n-126)*x+(729*n^2-2025*n+1134)*x^2-(729*n^2-1863*n+1008)*x^3)*d/dx P(n-2,x) + (117*n-190+(162*n^2-576*n+512)*x)*P(n-1,x) + ((81*n-144)*x-(81*n-144)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(198*n - 154 + (324*n^2 - 1152*n + 700)*x + (729*n^3 - 3888*n^2 + 6183*n - 2800)*x^2)*P[n - 2, x] - ((162*n - 126)*x + (729*n^2 - 2025*n + 1134)*x^2 - (729*n^2 - 1863*n + 1008)*x^3)*D[P[n - 2, x], x] + (117*n - 190 + (162*n^2 - 576*n + 512)*x)*P[n - 1, x] + ((81*n - 144)*x - (81*n - 144)*x^2)*D[P[n - 1, x], x])/(9*n - 16)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(198*n - 154 + (324*n^2 - 1152*n + 700)*x + (729*n^3 - 3888*n^2 + 6183*n - 2800)*x^2)*v[i - 2] - ((162*n - 126)*x + (729*n^2 - 2025*n + 1134)*x^2 - (729*n^2 - 1863*n + 1008)*x^3)*deriv(v[i - 2], x) + (117*n - 190 + (162*n^2 - 576*n + 512)*x)*v[i - 1] + ((81*n - 144)*x - (81*n - 144)*x^2)*deriv(v[i - 1], x))/(9*n - 16)); 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"],"edges":["e-9cba6f983f1ee78e"],"text":"A257619 \u2192 D"},"headline":"A257619 \u2192 D","chain":null}