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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":"e08c7d9a844b35911e2da9886a35c096895f07c0a548f6ca0bc113d369e8d718","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-05b91207a1ba3acf","role":"sequence","name":"Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).","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":[{"id":"l-85164ac6ca8b752c","kind":"fingerprint_related","node_a_id":"n-c8e1d395ac5ec819","node_b_id":"n-fafdfaf5597dbd80","fingerprint":"f6-01b9b2797a61b9d5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01b9b2797a61b9d5"},"other_id":"n-c8e1d395ac5ec819","other":{"id":"n-c8e1d395ac5ec819","name":"derivative of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","first_index":0,"fingerprint":"f6-01b9b2797a61b9d5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["10","20","3"],["20","100","60","4"]],"row_sums_preview":["0","1","6","33","184","1055","6222","37639"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-047228a38014e6d4","steps":["D"],"edges":["e-7c51df848162a1ed"],"text":"A056939 \u2192 D"},"headline":"A056939 \u2192 D","chain":null}}],"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:    120,       60;\n   2:    600,     1200,       180;\n   3:   2100,    10500,      6300,        420;\n   4:   5880,    58800,     88200,      23520,         840;\n   5:  14112,   246960,    740880,     493920,       70560,        1512;\n   6:  30240,   846720,   4445280,    5927040,     2116800,      181440,        2520;\n   7:  59400,  2494800,  20956320,   48898080,    34927200,     7484400,      415800,       3960;\n   8: 108900,  6534000,  82328400,  307359360,   384199200,   164656800,    22869000,     871200,      5940;\n   9: 188760, 15572700, 280308600, 1569728160,  3139456320,  2354592240,   654053400,   62290800,   1698840,    8580;\n  10: 312312, 34354320, 850269420, 6802155360, 20406466080, 24487759296, 11903771880, 2267385120, 154594440, 3123120, 12012;\n","recurrence_sentence":"The row polynomials satisfy (n^3+6*n^2+5*n)*P(n,x) = (n^3+12*n^2+47*n+60+(10*n^3+30*n^2+20*n)*x)*P(n-1,x) + ((3*n^2+39*n+60)*x-(15*n^2+27*n)*x^2)*d/dx P(n-1,x) + ((6*n+10)*x^2+(6*n+10)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 120 + 60*x.","mathematica":"P[1, x_] := 120 + 60*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 12*n^2 + 47*n + 60 + (10*n^3 + 30*n^2 + 20*n)*x)*P[n - 1, x] + ((3*n^2 + 39*n + 60)*x - (15*n^2 + 27*n)*x^2)*D[P[n - 1, x], x] + ((6*n + 10)*x^2 + (6*n + 10)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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