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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"1d2272615c573d01972d2b7fd774da9c42cc06b3d1bd2898a4ca40d644db7df6","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-dfe248045567f938","role":"sequence","name":"basis_map_A013609 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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:       1;\n   2:       3,       16;\n   3:      13,      144,       216;\n   4:      63,     1056,      3240,       2560;\n   5:     321,     7200,     33480,      53760,       28000;\n   6:    1683,    47280,    294840,     716800,      756000,      290304;\n   7:    8989,   303408,   2373840,    7741440,    12348000,     9580032,     2897664;\n   8:   48639,  1916992,  18044208,   73866240,   158004000,   185213952,   113008896,    28114944;\n   9:  265729, 11976768, 131770800,  649313280,  1743588000,  2739889152,  2523865344,  1265172480,   266872320;\n  10: 1462563, 74195280, 934266960, 5383741440, 17405388000, 34331641344, 42378336000, 32051036160, 13610488320, 2489344000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4+(9*n-9)*x-(20*n^3-80*n^2+99*n-39)*x^2-(24*n^3-96*n^2+126*n-54)*x^3+(4*n^5-28*n^4+76*n^3-100*n^2+64*n-16)*x^4)*P(n,x) = (12*n-12+(28*n^2+7*n-35)*x-(60*n^3-252*n^2+189*n+3)*x^2-(16*n^4+112*n^3-480*n^2+418*n-66)*x^3+(12*n^5-84*n^4+60*n^3+204*n^2-240*n+48)*x^4+(16*n^5-80*n^4+128*n^3-64*n^2)*x^5)*P(n-1,x) + (2+18*x-(28*n^2-56*n-38)*x^2-(132*n^2-264*n+6)*x^3+(8*n^4-32*n^3-168*n^2+400*n-76)*x^4+(24*n^4-96*n^3+192*n-48)*x^5+(16*n^4-64*n^3+64*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n - 12 + (28*n^2 + 7*n - 35)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (16*n^4 + 112*n^3 - 480*n^2 + 418*n - 66)*x^3 + (12*n^5 - 84*n^4 + 60*n^3 + 204*n^2 - 240*n + 48)*x^4 + (16*n^5 - 80*n^4 + 128*n^3 - 64*n^2)*x^5)*P[n - 1, x] + (2 + 18*x - (28*n^2 - 56*n - 38)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (8*n^4 - 32*n^3 - 168*n^2 + 400*n - 76)*x^4 + (24*n^4 - 96*n^3 + 192*n - 48)*x^5 + (16*n^4 - 64*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(4*n - 4 + (9*n - 9)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (24*n^3 - 96*n^2 + 126*n - 54)*x^3 + (4*n^5 - 28*n^4 + 76*n^3 - 100*n^2 + 64*n - 16)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((12*n - 12 + (28*n^2 + 7*n - 35)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (16*n^4 + 112*n^3 - 480*n^2 + 418*n - 66)*x^3 + (12*n^5 - 84*n^4 + 60*n^3 + 204*n^2 - 240*n + 48)*x^4 + (16*n^5 - 80*n^4 + 128*n^3 - 64*n^2)*x^5)*v[i - 1] + (2 + 18*x - (28*n^2 - 56*n - 38)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (8*n^4 - 32*n^3 - 168*n^2 + 400*n - 76)*x^4 + (24*n^4 - 96*n^3 + 192*n - 48)*x^5 + (16*n^4 - 64*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(4*n - 4 + (9*n - 9)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (24*n^3 - 96*n^2 + 126*n - 54)*x^3 + (4*n^5 - 28*n^4 + 76*n^3 - 100*n^2 + 64*n - 16)*x^4)); 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-4689fb1386d440ad","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-7c694c7111620c7d","e-0178d06b499a08b2"],"text":"A063007 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A063007 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}