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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":"fe25dd15b80cea3e36b83e09ad5c24e09e9b3c12416d7bf8267277b08ae550ee","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-5b112963159320a9","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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,       12;\n   3:     11,       96,        81;\n   4:     45,      612,      1215,       432;\n   5:    197,     3648,     11826,     10368,       2025;\n   6:    903,    21180,     96390,    142560,      70875,       8748;\n   7:   4279,   121536,    715635,   1520640,    1306125,     419904,      35721;\n   8:  20793,   693588,   5023053,  14046480,   17647875,    9981468,    2250423,    139968;\n   9: 103049,  3948288,  33981444, 118395648,  198478350,  166561920,   67012596,  11197440,   531441;\n  10: 518859, 22453740, 224050860, 937137600, 1978688250, 2240870184, 1353254364, 408146688, 52612659, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (12*n+12-(90*n^2+90*n)*x-(162*n^3-324*n^2-378*n+108)*x^2+(486*n^4-324*n^3-648*n^2+162*n)*x^3+(243*n^5-243*n^4-243*n^3+243*n^2)*x^4)*P(n,x) = (60*n-12-(324*n^2-144*n+180)*x-(1134*n^3-324*n^2-1566*n-108)*x^2+(2106*n^4-3159*n^3+2592*n^2-81*n-2106)*x^3+(1215*n^5+4617*n^4-8991*n^3-1701*n^2+4860*n)*x^4+(2187*n^5-729*n^4-4374*n^3+2187*n^2+2187*n-1458)*x^5)*P(n-1,x) + (2-36*n*x-(54*n^2-45*n-27)*x^2+(324*n^3-54*n^2+162*n)*x^3+(162*n^4+162*n^3+324*n^2-486*n-324)*x^4-(2916*n^3-486*n^2-1458*n)*x^5-(1458*n^4+1458*n^3-1458*n^2-729*n+729)*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[((60*n - 12 - (324*n^2 - 144*n + 180)*x - (1134*n^3 - 324*n^2 - 1566*n - 108)*x^2 + (2106*n^4 - 3159*n^3 + 2592*n^2 - 81*n - 2106)*x^3 + (1215*n^5 + 4617*n^4 - 8991*n^3 - 1701*n^2 + 4860*n)*x^4 + (2187*n^5 - 729*n^4 - 4374*n^3 + 2187*n^2 + 2187*n - 1458)*x^5)*P[n - 1, x] + (2 - 36*n*x - (54*n^2 - 45*n - 27)*x^2 + (324*n^3 - 54*n^2 + 162*n)*x^3 + (162*n^4 + 162*n^3 + 324*n^2 - 486*n - 324)*x^4 - (2916*n^3 - 486*n^2 - 1458*n)*x^5 - (1458*n^4 + 1458*n^3 - 1458*n^2 - 729*n + 729)*x^6)*D[P[n - 1, x], x])/(12*n + 12 - (90*n^2 + 90*n)*x - (162*n^3 - 324*n^2 - 378*n + 108)*x^2 + (486*n^4 - 324*n^3 - 648*n^2 + 162*n)*x^3 + (243*n^5 - 243*n^4 - 243*n^3 + 243*n^2)*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] = ((60*n - 12 - (324*n^2 - 144*n + 180)*x - (1134*n^3 - 324*n^2 - 1566*n - 108)*x^2 + (2106*n^4 - 3159*n^3 + 2592*n^2 - 81*n - 2106)*x^3 + (1215*n^5 + 4617*n^4 - 8991*n^3 - 1701*n^2 + 4860*n)*x^4 + (2187*n^5 - 729*n^4 - 4374*n^3 + 2187*n^2 + 2187*n - 1458)*x^5)*v[i - 1] + (2 - 36*n*x - (54*n^2 - 45*n - 27)*x^2 + (324*n^3 - 54*n^2 + 162*n)*x^3 + (162*n^4 + 162*n^3 + 324*n^2 - 486*n - 324)*x^4 - (2916*n^3 - 486*n^2 - 1458*n)*x^5 - (1458*n^4 + 1458*n^3 - 1458*n^2 - 729*n + 729)*x^6)*deriv(v[i - 1], x))/(12*n + 12 - (90*n^2 + 90*n)*x - (162*n^3 - 324*n^2 - 378*n + 108)*x^2 + (486*n^4 - 324*n^3 - 648*n^2 + 162*n)*x^3 + (243*n^5 - 243*n^4 - 243*n^3 + 243*n^2)*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-0d75f6272d2a2a84","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-edd649892838aaa7","e-21413828b88c663d"],"text":"A114656 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A114656 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}