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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":"d379d9b2840840caef6c489379bf1e139cc473b95f984b4d8b2d62779174836b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cbba416b5dae7a51","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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:      3,       6;\n  2:     11,      42,       27;\n  3:     45,     252,      324,       108;\n  4:    197,    1452,     2808,      1944,       405;\n  5:    903,    8250,    21330,     22680,     10125,      1458;\n  6:   4279,   46638,   151065,    217620,    151875,     48114,     5103;\n  7:  20793,  263256,  1025136,   1867320,   1786050,    898128,   214326,    17496;\n  8: 103049, 1486104,  6761664,  14908320,  18178020,  12655440,  4858056,   909792,   59049;\n  9: 518859, 8395758, 43711812, 113200416, 168092820, 150273144, 80586576, 24564384, 3720087, 196830;\n","recurrence_sentence":"The row polynomials satisfy (12*n+24+(36*n^2+108*n+72)*x+(27*n^3+81*n^2+54*n)*x^2)*P(n,x) = (84*n+24+(288*n^2+504*n+72)*x+(189*n^3+891*n^2+702*n)*x^2+(243*n^3+567*n^2+162*n)*x^3)*P(n-1,x) - (8+(48*n+84)*x+(36*n^2+306*n+288)*x^2+(162*n^2+621*n+378)*x^3+(162*n^2+405*n+162)*x^4)*d/dx 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[((84*n + 24 + (288*n^2 + 504*n + 72)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (243*n^3 + 567*n^2 + 162*n)*x^3)*P[n - 1, x] - (8 + (48*n + 84)*x + (36*n^2 + 306*n + 288)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (162*n^2 + 405*n + 162)*x^4)*D[P[n - 1, x], x])/(12*n + 24 + (36*n^2 + 108*n + 72)*x + (27*n^3 + 81*n^2 + 54*n)*x^2)]];\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] = ((84*n + 24 + (288*n^2 + 504*n + 72)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (243*n^3 + 567*n^2 + 162*n)*x^3)*v[i - 1] - (8 + (48*n + 84)*x + (36*n^2 + 306*n + 288)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (162*n^2 + 405*n + 162)*x^4)*deriv(v[i - 1], x))/(12*n + 24 + (36*n^2 + 108*n + 72)*x + (27*n^3 + 81*n^2 + 54*n)*x^2)); 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-57b4b14604f4c0d7","steps":["basis A013610","\u03b8+1"],"edges":["e-125d456539fb62ba","e-140d32720595d668"],"text":"A126216 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A126216 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}