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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":"5f7cf0dec62e532c90d852cb25d8f061bae61a075a305431959fb4dacc2b77e9","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-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 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:       25,       22,         4;\n  1:      117,      164,        68,         8;\n  2:      525,     1020,       684,       184,        16;\n  3:     2289,     5710,      5356,      2336,       464,        32;\n  4:     9773,    29834,     36152,     22112,      7120,      1120,       64;\n  5:    41065,   148456,    221112,    175120,     79120,     20160,     2624,      128;\n  6:   170393,   712344,   1260552,   1229552,    719920,    257344,    54208,     6016,     256;\n  7:   699869,  3323130,   6817416,   7914016,   5711984,   2646560,   781760,   140288,   13568,    512;\n  8:  2850625, 15158110,  35389404,  47694944,  41004656,  23427936,  8968960,  2255872,  352512,  30208,  1024;\n  9: 11529437, 67885004, 177777308, 272997208, 272705840, 185674496, 87659264, 28568064, 6253824, 865280, 66560, 2048;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+9*n^2+13*n+6-(4*n+6)*x-(2*n+3)*x^2)*P(n,x) = (4*n^3+27*n^2+53*n+30+(2*n^3+14*n^2+16*n-6)*x-(10*n+19)*x^2-(2*n+4)*x^3)*P(n-1,x) + (6*n^2+24*n+(8*n^2+32*n-24)*x+(2*n^2+8*n-38)*x^2-16*x^3-2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 25 + 22*x + 4*x^2.","mathematica":"P[0, x_] := 25 + 22*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 27*n^2 + 53*n + 30 + (2*n^3 + 14*n^2 + 16*n - 6)*x - (10*n + 19)*x^2 - (2*n + 4)*x^3)*P[n - 1, x] + (6*n^2 + 24*n + (8*n^2 + 32*n - 24)*x + (2*n^2 + 8*n - 38)*x^2 - 16*x^3 - 2*x^4)*D[P[n - 1, x], x])/(2*n^3 + 9*n^2 + 13*n + 6 - (4*n + 6)*x - (2*n + 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 25 + 22*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 27*n^2 + 53*n + 30 + (2*n^3 + 14*n^2 + 16*n - 6)*x - (10*n + 19)*x^2 - (2*n + 4)*x^3)*v[i - 1] + (6*n^2 + 24*n + (8*n^2 + 32*n - 24)*x + (2*n^2 + 8*n - 38)*x^2 - 16*x^3 - 2*x^4)*deriv(v[i - 1], x))/(2*n^3 + 9*n^2 + 13*n + 6 - (4*n + 6)*x - (2*n + 3)*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 A007318","D+1"],"edges":["e-1acefd6886502727","e-6d9279bdfa8e0bba"],"text":"A123519 \u2192 basis A007318 \u2192 D+1"},"headline":"A123519 \u2192 basis A007318 \u2192 D+1","chain":null}