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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":"76fe70c2f3836dc62fd8a280456ba11e579530c6bcc2601b3bca81cca904f02a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b0592fccca597966","role":"sequence","name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 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:         1,          12;\n  2:         4,          60,          81;\n  3:        28,         468,         972,         432;\n  4:       280,        5016,       12879,        9504,        2025;\n  5:      3640,       68568,      201285,      192240,       70875,        8748;\n  6:     58240,     1140768,     3683394,     4149360,     2035125,      446148,      35721;\n  7:   1106560,    22373472,    77684670,    98482608,    58117500,    17268552,    2500470,    139968;\n  8:  24344320,   505495104,  1860083676,  2580935616,  1740222225,   630975744,  125523594,  12877056,   531441;\n  9: 608608000, 12929509440, 49914183852, 74443836672, 55603691325, 23292153612, 5714574138, 813773952, 62178597, 1968300;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(36*n-60)*x-81*x^2)*P(n-2,x) + ((36*n-60)*x^2+36*x^3)*d/dx P(n-2,x) + (6*n-7+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (36*n - 60)*x - 81*x^2)*P[n - 2, x] + ((36*n - 60)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (6*n - 7 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (36*n - 60)*x - 81*x^2)*v[i - 2] + ((36*n - 60)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (6*n - 7 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); 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-1a0f876fa1441737","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-6e846af4747cc041","e-ad74633e7b3aac25"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}