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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":"c9a81b74b184d2964fc432669088d1a30a5a293da60198ef49ee7ce853802539","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-4d9510ff3052f23c","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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,             3;\n  2:           38,            84,             27;\n  3:          410,          1812,           2187,             486;\n  4:         6190,         42852,         100287,           85536,           14580;\n  5:       126760,       1215012,        4371687,         7106211,         4738500,           656100;\n  6:      3409648,      42081096,      207313272,       512028216,       643792293,        354294000,         41334300;\n  7:    117093680,    1774958760,    11180204712,     37523468808,     71117855253,      73281287412,      34431471900,       3472081200;\n  8:   5011583984,   90286591176,   695227393656,   2968363066680,   7603895780469,   11744281574064,   10282049078796,    4222050739200,    374984769600;\n  9: 261942062624, 5474901670416, 49940917502076, 259761709396980, 843350145857289, 1753499683652139, 2291575805392716, 1747284258264660, 637849093089600, 50622943896000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(6*n^5-6*n^4+6*n^3+6*n^2-12*n)*x+(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(12*n^3-12*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (12*n^3 - 12*n)*x)*P[n - 1, x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (12*n^3 - 12*n)*x)*v[i - 1])/(4*n - 4)); 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-ecd77b030b8fab94","steps":["basis A013610","D+1"],"edges":["e-201bedfe8f6594c1","e-18aad98ed0ac7858"],"text":"A128813 \u2192 basis A013610 \u2192 D+1"},"headline":"A128813 \u2192 basis A013610 \u2192 D+1","chain":null}