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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":"66ae2a2944864e8e612e0ab1eb61ec4c818e27d3d704bb188125c88ff2b16cee","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:        1;\n  1:        2,           1;\n  2:       19,          20,            1;\n  3:      180,         537,          182,             1;\n  4:     1637,       13808,        10994,          1640,             1;\n  5:    14758,      303481,       577448,        203390,         14762,             1;\n  6:   132855,     6015124,     24595435,      18982640,       3605405,        132860,            1;\n  7:  1195736,   112088353,    887976174,    1387549775,     554741500,      62620719,      1195742,           1;\n  8: 10761673,  2013118208,  28634320900,   82473675664,   64370894750,   15218117152,   1076509972,    10761680,        1;\n  9: 96855114, 35363746833, 855635217056, 4207612448908, 5892417787764, 2658073173686, 402768044752, 18408286764, 96855122, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n-23+16*x)*P(n,x) = -(72*n-135+(128*n^2-672*n+954)*x+(512*n^3-3904*n^2+9560*n-7959)*x^2+(1536*n^2-7808*n+9368)*x^3+(1024*n-1920)*x^4)*P(n-2,x) - ((64*n-120)*x+(512*n^2-2496*n+3008)*x^2-(512*n^2-3968*n+6792)*x^3-(1536*n-4928)*x^4-1024*x^5)*d/dx P(n-2,x) + (80*n-222+(128*n^2-672*n+1034)*x+(320*n-856)*x^2+128*x^3)*P(n-1,x) + ((64*n-184)*x-(64*n-312)*x^2-128*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(72*n - 135 + (128*n^2 - 672*n + 954)*x + (512*n^3 - 3904*n^2 + 9560*n - 7959)*x^2 + (1536*n^2 - 7808*n + 9368)*x^3 + (1024*n - 1920)*x^4)*P[n - 2, x] - ((64*n - 120)*x + (512*n^2 - 2496*n + 3008)*x^2 - (512*n^2 - 3968*n + 6792)*x^3 - (1536*n - 4928)*x^4 - 1024*x^5)*D[P[n - 2, x], x] + (80*n - 222 + (128*n^2 - 672*n + 1034)*x + (320*n - 856)*x^2 + 128*x^3)*P[n - 1, x] + ((64*n - 184)*x - (64*n - 312)*x^2 - 128*x^3)*D[P[n - 1, x], x])/(8*n - 23 + 16*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] = (-(72*n - 135 + (128*n^2 - 672*n + 954)*x + (512*n^3 - 3904*n^2 + 9560*n - 7959)*x^2 + (1536*n^2 - 7808*n + 9368)*x^3 + (1024*n - 1920)*x^4)*v[i - 2] - ((64*n - 120)*x + (512*n^2 - 2496*n + 3008)*x^2 - (512*n^2 - 3968*n + 6792)*x^3 - (1536*n - 4928)*x^4 - 1024*x^5)*deriv(v[i - 2], x) + (80*n - 222 + (128*n^2 - 672*n + 1034)*x + (320*n - 856)*x^2 + 128*x^3)*v[i - 1] + ((64*n - 184)*x - (64*n - 312)*x^2 - 128*x^3)*deriv(v[i - 1], x))/(8*n - 23 + 16*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-90af30f849fc554d","steps":["D+1"],"edges":["e-2ddf9f1f4c4ecfaa"],"text":"A167884 \u2192 D+1"},"headline":"A167884 \u2192 D+1","chain":null}