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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":"77721e90a6d54ec7e8ce521e2903a655b7caed6ecdb6ca7662d2542f7a498db6","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-8401ba73e6a78cca","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:          5,           2;\n  2:         37,          38,            8;\n  3:        349,         620,          332,           48;\n  4:       3981,       10324,         9452,         3472,          384;\n  5:      53241,      183734,       243652,       152072,        42784,         3840;\n  6:     816873,     3541586,      6217612,      5593592,      2664064,       609408,       46080;\n  7:   14144985,    74104344,    162865320,    193292992,    132263184,     51099008,     9868032,      645120;\n  8:  272897145,  1679980968,   4450271016,   6595144000,   5938504464,   3288667776,  1070975744,   179177472,   10321920;\n  9: 5805315405, 41123665962, 127768695960, 227789610640, 255628075728, 185939604384, 86650256384, 24420773376, 3607105536, 185794560;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(8*n^3-12*n^2+4*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(n - 1)); 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-7fd16a32c438d522","steps":["basis A013609","D+1"],"edges":["e-23e30432bbaed1be","e-425807e42cd0e364"],"text":"A059364 \u2192 basis A013609 \u2192 D+1"},"headline":"A059364 \u2192 basis A013609 \u2192 D+1","chain":null}