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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":"b006944883938465454a0a046e1d2e77c2e094abf6e9acca2bdc228128f7a356","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-623036dc25b33b8c","role":"sequence","name":"basis_map_A013609 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:            4,             2;\n  2:           28,            44,             12;\n  3:          292,           884,            756,            144;\n  4:         4332,         20284,          32476,          19584,            2880;\n  5:        87792,        565984,        1376656,        1529544,          720000,           86400;\n  6:      2345376,      19412224,       64306768,      107326368,        92176848,        35769600,         3628800;\n  7:     80158496,     813494016,     3435923856,     7752722336,      9926204688,      6983042688,      2311545600,      203212800;\n  8:   3418609568,   41189356800,   212327203152,   607823932832,   1046617911504,   1091766198528,    652318292736,   188581478400,    14631321600;\n  9: 178195506368, 2489120668800, 15184246815552, 52869114168992, 115074675129024, 160810760125728, 141909775151616, 73821175856640, 18962192793600, 1316818944000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(4*n^5-4*n^4+4*n^3+4*n^2-8*n)*x+(4*n^5-4*n^4-4*n^3+4*n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(8*n^3-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (8*n^3 - 8*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (8*n^3 - 8*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 A013609","D+1"],"edges":["e-42bb566932c0b2ac","e-ea6fc32cb558cd6d"],"text":"A128813 \u2192 basis A013609 \u2192 D+1"},"headline":"A128813 \u2192 basis A013609 \u2192 D+1","chain":null}