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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":"71a7384987eb552e5f79eec3f64bf2d4d22ab6f29e486d1c5798992502ff375c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2b620f031a59b07c","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:          12,            13,             1;\n   2:         112,           171,            33,             1;\n   3:        1112,          2473,           855,            62,             1;\n   4:       11112,         39211,         20965,          2685,           100,            1;\n   5:      111112,        667233,        513831,        101870,          6525,          147,           1;\n   6:     1111112,      11899651,      12830013,       3654301,        355460,        13482,         203,         1;\n   7:    11111112,     218315593,     327985855,     128210502,      17654021,       998522,       24906,       268,        1;\n   8:   111111112,    4070218491,    8573101605,    4473418125,     833463540,     65692683,     2407818,     42390,      342,      1;\n   9:  1111111112,   76556373553,  228280633911,  156506777790,   38261603325,   4023246969,   202755147,   5184480,    67770,    425,   1;\n  10: 11111111112, 1446793319731, 6166633073293, 5510876738301, 1730565643620, 235516598724, 15639178401, 543844587, 10229340, 103125, 517, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10-7*x-8*x^2)*P(n-2,x) - (9*x-72*x^2)*d/dx P(n-2,x) + (11-7*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 13*x + x^2, P(2,x) = 112 + 171*x + 33*x^2 + x^3.","mathematica":"P[1, x_] := 12 + 13*x + x^2;\nP[2, x_] := 112 + 171*x + 33*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(10 - 7*x - 8*x^2)*P[n - 2, x] - (9*x - 72*x^2)*D[P[n - 2, x], x] + (11 - 7*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 12 + 13*x + x^2; v[2] = 112 + 171*x + 33*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = -(10 - 7*x - 8*x^2)*v[i - 2] - (9*x - 72*x^2)*deriv(v[i - 2], x) + (11 - 7*x)*v[i - 1] + 9*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2b620f031a59b07c","steps":["D+1"],"edges":["e-5301fd580d089957"],"text":"A166979 \u2192 D+1"},"headline":"A166979 \u2192 D+1","chain":null}