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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":"a324916c77aca6a9be58af634e0a1dbd1dd07842a645a30a5bee0c731c006662","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-51fcbaaf07b9691b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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:    2,      6;\n   3:    6,     30,      27;\n   4:   16,    120,     216,     108;\n   5:   44,    432,    1161,    1188,     405;\n   6:  120,   1464,    5238,    8100,    5670,     1458;\n   7:  328,   4776,   21330,   44172,   46980,    24786,    5103;\n   8:  896,  15168,   81216,  210816,  301320,   242028,  102060,   17496;\n   9: 2448,  47232,  294840,  920160, 1652805,  1787508, 1148175,  402408,   59049;\n  10: 6688, 144864, 1032912, 3766176, 8149410, 11093922, 9603846, 5126328, 1535274, 196830;\n","recurrence_sentence":"The row polynomials satisfy (16*n-16+(12*n^2+8*n-20)*x+(3*n^3+6*n^2-9*n)*x^2)*P(n,x) = -(32*n-96+(36*n^2-92*n-144)*x+(6*n^3-162*n-48)*x^2-(18*n^2+54*n)*x^3)*P(n-1,x) + (32+(48*n+96)*x+(12*n^2+132*n+88)*x^2+(24*n^2+108*n+24)*x^3+(9*n^2+27*n)*x^4)*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[(-(32*n - 96 + (36*n^2 - 92*n - 144)*x + (6*n^3 - 162*n - 48)*x^2 - (18*n^2 + 54*n)*x^3)*P[n - 1, x] + (32 + (48*n + 96)*x + (12*n^2 + 132*n + 88)*x^2 + (24*n^2 + 108*n + 24)*x^3 + (9*n^2 + 27*n)*x^4)*D[P[n - 1, x], x])/(16*n - 16 + (12*n^2 + 8*n - 20)*x + (3*n^3 + 6*n^2 - 9*n)*x^2)]];\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] = (-(32*n - 96 + (36*n^2 - 92*n - 144)*x + (6*n^3 - 162*n - 48)*x^2 - (18*n^2 + 54*n)*x^3)*v[i - 1] + (32 + (48*n + 96)*x + (12*n^2 + 132*n + 88)*x^2 + (24*n^2 + 108*n + 24)*x^3 + (9*n^2 + 27*n)*x^4)*deriv(v[i - 1], x))/(16*n - 16 + (12*n^2 + 8*n - 20)*x + (3*n^3 + 6*n^2 - 9*n)*x^2)); 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-c72653d9ab0ee8be","steps":["basis A013610","\u03b8+1"],"edges":["e-52226b414d15e69f","e-2422fa5edfa449a1"],"text":"A063967 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A063967 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}