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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":"ac280c0d05f992c6b29f40db364f7541bb79d1d7fd60eb8bb6bf9c36affc1f30","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-07c6a4edd51718c6","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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:       67,       184,         64;\n  1:      345,      1712,       2240,         512;\n  2:     1641,     11856,      28224,       24064,         4096;\n  3:     7443,     70744,     246976,      382976,       241664,        32768;\n  4:    32699,    385640,    1781120,     4057088,      4648960,      2326528,        262144;\n  5:   140401,   1977952,   11382912,    34370560,     57856000,     52494336,      21757952,       2097152;\n  6:   592529,   9708960,   66956160,   252382208,    563507200,    750714880,     563347456,     199229440,      16777216;\n  7:  2467163,  46100904,  370697856,  1677162496,   4678406144,   8267399168,    9103474688,    5821693952,    1795162112,    134217728;\n  8: 10162291, 213283096, 1959620544, 10352052224,  34689191936,  76607225856,  111869427712,  104907931648,   58435043328,  15971909632,   1073741824;\n  9: 41489225, 966320912, 9988185920, 60362286592, 236360314880, 627284639744, 1144773410816, 1423253176320, 1161805430784, 573109698560, 140660178944, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy (64*n^3+288*n^2+452*n+246+(64*n+96)*x-(64*n+96)*x^2)*P(n,x) = (128*n^3+864*n^2+1912*n+1374+(256*n^3+1792*n^2+4112*n+3072)*x-(128*n+224)*x^2-(256*n+512)*x^3)*P(n-1,x) + (48*n^2+192*n+171+(256*n^2+1024*n+864)*x+(256*n^2+1024*n+608)*x^2-512*x^3-256*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 67 + 184*x + 64*x^2.","mathematica":"P[0, x_] := 67 + 184*x + 64*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((128*n^3 + 864*n^2 + 1912*n + 1374 + (256*n^3 + 1792*n^2 + 4112*n + 3072)*x - (128*n + 224)*x^2 - (256*n + 512)*x^3)*P[n - 1, x] + (48*n^2 + 192*n + 171 + (256*n^2 + 1024*n + 864)*x + (256*n^2 + 1024*n + 608)*x^2 - 512*x^3 - 256*x^4)*D[P[n - 1, x], x])/(64*n^3 + 288*n^2 + 452*n + 246 + (64*n + 96)*x - (64*n + 96)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 67 + 184*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((128*n^3 + 864*n^2 + 1912*n + 1374 + (256*n^3 + 1792*n^2 + 4112*n + 3072)*x - (128*n + 224)*x^2 - (256*n + 512)*x^3)*v[i - 1] + (48*n^2 + 192*n + 171 + (256*n^2 + 1024*n + 864)*x + (256*n^2 + 1024*n + 608)*x^2 - 512*x^3 - 256*x^4)*deriv(v[i - 1], x))/(64*n^3 + 288*n^2 + 452*n + 246 + (64*n + 96)*x - (64*n + 96)*x^2)); 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-8eb9bb3a09f0b7fd","steps":["basis A013611","D+1"],"edges":["e-ab96494d3bfaa8ca","e-1b6f7fa0d50055d6"],"text":"A123519 \u2192 basis A013611 \u2192 D+1"},"headline":"A123519 \u2192 basis A013611 \u2192 D+1","chain":null}