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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":"9f553b234faeb3ac0db474dfe2dcbe90822ee34a4d87ac18c13735f9b81cc89d","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-2117bc35fff39734","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","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:            1105,             1968,               576;\n  1:           25721,            79560,             70848,              13824;\n  2:          555169,          2469216,           3818880,            2267136,             331776;\n  3:        11442377,         66620280,         148152960,          152755200,           68014080,            7962624;\n  4:       228596977,       1647702288,        4796660160,         7111342080,         5499187200,         1958805504,           191102976;\n  5:      4465450265,      38404292136,      138406992192,       268612968960,       298676367360,       184772689920,         54846554112,         4586471424;\n  6:     85760782657,     857366450880,     3686812045056,      8858047500288,     12893422510080,     11469172506624,       5912726077440,      1504362627072,       110075314176;\n  7:   1625351483753,   18524329053912,    92595576695040,    265450467244032,    478334565015552,    556995852435456,     412890210238464,    182449833246720,     40617790930944,      2641807540224;\n  8:  30477084259729,  390084356100720,  2222919486469440,   7407646135603200,  15927028034641920,  22960059120746496,   22279834097418240,  14156235779604480,   5473494997401600,   1083141091491840,    63403380965376;\n  9: 566494286026169, 8045950244568456, 51491135005295040, 195616914809310720, 488904644949811200, 840947080229093376, 1010242601312845824, 840268028816916480, 467155780726947840, 160555853257113600, 28594924815384576, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy (24*n+41+24*x)*P(n,x) = (408*n+1105+(576*n+1968)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1105 + 1968*x + 576*x^2.","mathematica":"P[0, x_] := 1105 + 1968*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((408*n + 1105 + (576*n + 1968)*x + 576*x^2)*P[n - 1, x])/(24*n + 41 + 24*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1105 + 1968*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((408*n + 1105 + (576*n + 1968)*x + 576*x^2)*v[i - 1])/(24*n + 41 + 24*x)); 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-1f2dec04776afb2b","steps":["basis A013611","D+1"],"edges":["e-5fcdd9bd1fb647dd","e-bf0a2d02672a2068"],"text":"A038320 \u2192 basis A013611 \u2192 D+1"},"headline":"A038320 \u2192 basis A013611 \u2192 D+1","chain":null}