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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":"4a8cceb6b515cdd91fd5f81459dce5d0819d6c84ffccaf770bf37d4fbbe43c74","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-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) 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 = 6.","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,          1;\n  2:       15,         16,            1;\n  3:      112,        333,          114,            1;\n  4:      797,       6568,         5274,          800,            1;\n  5:     5598,     110881,       210568,        75030,         5602,            1;\n  6:    39211,    1687752,      6825459,      5297480,      1020321,        39216,           1;\n  7:   274508,   24136573,   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- 792)*x^4)*P[n - 2, x] - ((36*n - 66)*x + (216*n^2 - 1044*n + 1260)*x^2 - (216*n^2 - 1656*n + 2814)*x^3 - (648*n - 2052)*x^4 - 432*x^5)*D[P[n - 2, x], x] + (48*n - 130 + (72*n^2 - 372*n + 572)*x + (180*n - 474)*x^2 + 72*x^3)*P[n - 1, x] + ((36*n - 102)*x - (36*n - 174)*x^2 - 72*x^3)*D[P[n - 1, x], x])/(6*n - 17 + 12*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(42*n - 77 + (72*n^2 - 372*n + 524)*x + (216*n^3 - 1620*n^2 + 3942*n - 3329)*x^2 + (648*n^2 - 3240*n + 3834)*x^3 + (432*n - 792)*x^4)*v[i - 2] - ((36*n - 66)*x + (216*n^2 - 1044*n + 1260)*x^2 - (216*n^2 - 1656*n + 2814)*x^3 - (648*n - 2052)*x^4 - 432*x^5)*deriv(v[i - 2], x) + (48*n - 130 + (72*n^2 - 372*n + 572)*x + (180*n - 474)*x^2 + 72*x^3)*v[i - 1] + ((36*n - 102)*x - (36*n - 174)*x^2 - 72*x^3)*deriv(v[i - 1], x))/(6*n - 17 + 12*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-3bc26d3108ee6bfd","steps":["D+1"],"edges":["e-6b443a04a9d8189c"],"text":"A142461 \u2192 D+1"},"headline":"A142461 \u2192 D+1","chain":null}