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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":"e285e051e41443b47ce57d43b9c2da8352b8640089c9d9c00bb9cd55b676065f","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-587b2e7eeca5ddbb","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:             1;\n  1:            43,             24;\n  2:          1273,           2064,             576;\n  3:         32851,          91656,           74304,           13824;\n  4:        788785,        3153696,         4399488,         2377728,          331776;\n  5:      18114619,       94654200,       189221760,       175979520,        71331840,         7962624;\n  6:     403604137,     2608505136,      6815102400,      9082644480,      6335262720,      2054356992,       191102976;\n  7:    8797579747,    67805495016,    219114431424,    381645734400,    381471068160,    212864827392,     57521995776,      4586471424;\n  8:  188606936929,  1689135311424,   6509327521536,  14023323611136,  18318995251200,  14648489017344,   6811674476544,   1577746169856,   110075314176;\n  9: 3991137314635, 40739098376664, 182426613633792, 468671581550592, 757259475001344, 791380594851840, 527345604624384, 210188812419072, 42599146586112, 2641807540224;\n","recurrence_sentence":"The row polynomials satisfy (24*n-5+24*x)*P(n,x) = (456*n+361+(576*n+912)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((456*n + 361 + (576*n + 912)*x + 576*x^2)*P[n - 1, x])/(24*n - 5 + 24*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((456*n + 361 + (576*n + 912)*x + 576*x^2)*v[i - 1])/(24*n - 5 + 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-2f02b2035f8f4541","steps":["basis A013610","D+1"],"edges":["e-f9ff39da4c6bf2a9","e-b0b10fd71372f12e"],"text":"A038322 \u2192 basis A013610 \u2192 D+1"},"headline":"A038322 \u2192 basis A013610 \u2192 D+1","chain":null}