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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":"518822474d2414ff659b0da44d545fbd7775a8dad9e31e0dcd475a6f038c2ca7","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-98aa86c8bf356480","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:      46,       69,       18;\n  1:     166,      438,      315,        54;\n  2:     554,     2139,     2700,      1269,       162;\n  3:    1763,     9036,    16713,     13716,      4779,       486;\n  4:    5434,    34848,    85806,    102978,     62370,     17253,      1458;\n  5:   16369,   126282,   390285,    623430,    550800,    263898,     60507,      4374;\n  6:   48464,   437421,  1631160,   3272697,   3844260,   2680533,   1061424,    207765,     13122;\n  7:  141566,  1464222,  6405165,  15522732,  22895946,  21197862,  12191067,   4111560,    702027,    39366;\n  8:  409072,  4771803, 23977404,  68249358, 121619232, 141296967, 107754948,  52689204,  15470838,  2342277,  118098;\n  9: 1171621, 15219648, 86429538, 282939264, 592451415, 831373956, 794975958, 514828548, 218835594, 56883870, 7735419, 354294;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3-426*n-594+(324*n^3+1458*n^2+2304*n+1188)*x-54*n*x^2-(324*n+486)*x^3)*P(n,x) = (81*n^3+108*n^2-1116*n-2364+(567*n^3+3402*n^2+6030*n+2628)*x+(486*n^3+3402*n^2+6939*n+4374)*x^2-(1053*n+1782)*x^3-(486*n+972)*x^4)*P(n-1,x) + (45*n^2+45*n-220+(432*n^2+1242*n-177)*x+(1053*n^2+3969*n+1233)*x^2+(486*n^2+1944*n-1269)*x^3-2025*x^4-486*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 46 + 69*x + 18*x^2.","mathematica":"P[0, x_] := 46 + 69*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((81*n^3 + 108*n^2 - 1116*n - 2364 + (567*n^3 + 3402*n^2 + 6030*n + 2628)*x + (486*n^3 + 3402*n^2 + 6939*n + 4374)*x^2 - (1053*n + 1782)*x^3 - (486*n + 972)*x^4)*P[n - 1, x] + (45*n^2 + 45*n - 220 + (432*n^2 + 1242*n - 177)*x + (1053*n^2 + 3969*n + 1233)*x^2 + (486*n^2 + 1944*n - 1269)*x^3 - 2025*x^4 - 486*x^5)*D[P[n - 1, x], x])/(54*n^3 - 426*n - 594 + (324*n^3 + 1458*n^2 + 2304*n + 1188)*x - 54*n*x^2 - (324*n + 486)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 46 + 69*x + 18*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((81*n^3 + 108*n^2 - 1116*n - 2364 + (567*n^3 + 3402*n^2 + 6030*n + 2628)*x + (486*n^3 + 3402*n^2 + 6939*n + 4374)*x^2 - (1053*n + 1782)*x^3 - (486*n + 972)*x^4)*v[i - 1] + (45*n^2 + 45*n - 220 + (432*n^2 + 1242*n - 177)*x + (1053*n^2 + 3969*n + 1233)*x^2 + (486*n^2 + 1944*n - 1269)*x^3 - 2025*x^4 - 486*x^5)*deriv(v[i - 1], x))/(54*n^3 - 426*n - 594 + (324*n^3 + 1458*n^2 + 2304*n + 1188)*x - 54*n*x^2 - (324*n + 486)*x^3)); 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-dfab35176bd08e72","steps":["basis A013610","D+1"],"edges":["e-77ee9774535f66a4","e-fcd3c82fb53afbc4"],"text":"A207606 \u2192 basis A013610 \u2192 D+1"},"headline":"A207606 \u2192 basis A013610 \u2192 D+1","chain":null}