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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":"f8e33929ec1c4466ef28f9e947e730d673af6ab6e257c54365643ca8a84a1a43","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-90af30f849fc554d","role":"sequence","name":"Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","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:        1;\n  2:       18,           2;\n  3:      179,         358,            3;\n  4:     1636,       12172,         4908,             4;\n  5:    14757,      288724,       433086,         59028,             5;\n  6:   132854,     5882270,     21654300,      11764540,        664270,             6;\n  7:  1195735,   110892618,    832529865,    1110039820,     277231545,       7174410,            7;\n  8: 10761672,  2002356536,  27633142632,   73262628120,   46055237720,    6007069608,     75331704,         8;\n  9: 96855113, 35266891720, 838001771196, 3928278525176, 4910348156470, 1676003542392, 123434121020, 774840904, 9;\n","recurrence_sentence":"The row polynomials satisfy (8*n-15)*P(n,x) = -(72*n-63+(128*n^2-480*n+322)*x+(512*n^3-2880*n^2+4888*n-2415)*x^2)*P(n-2,x) - ((64*n-56)*x+(512*n^2-1472*n+896)*x^2-(512*n^2-1408*n+840)*x^3)*d/dx P(n-2,x) + (80*n-142+(128*n^2-480*n+450)*x)*P(n-1,x) + ((64*n-120)*x-(64*n-120)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(72*n - 63 + (128*n^2 - 480*n + 322)*x + (512*n^3 - 2880*n^2 + 4888*n - 2415)*x^2)*P[n - 2, x] - ((64*n - 56)*x + (512*n^2 - 1472*n + 896)*x^2 - (512*n^2 - 1408*n + 840)*x^3)*D[P[n - 2, x], x] + (80*n - 142 + (128*n^2 - 480*n + 450)*x)*P[n - 1, x] + ((64*n - 120)*x - (64*n - 120)*x^2)*D[P[n - 1, x], x])/(8*n - 15)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(72*n - 63 + (128*n^2 - 480*n + 322)*x + (512*n^3 - 2880*n^2 + 4888*n - 2415)*x^2)*v[i - 2] - ((64*n - 56)*x + (512*n^2 - 1472*n + 896)*x^2 - (512*n^2 - 1408*n + 840)*x^3)*deriv(v[i - 2], x) + (80*n - 142 + (128*n^2 - 480*n + 450)*x)*v[i - 1] + ((64*n - 120)*x - (64*n - 120)*x^2)*deriv(v[i - 1], x))/(8*n - 15)); 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-90af30f849fc554d","steps":["D"],"edges":["e-6b5939472eff07c8"],"text":"A167884 \u2192 D"},"headline":"A167884 \u2192 D","chain":null}