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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":"92fe9282673dee55c6a198b4f6cf19069a6cbc10ad360f8d18ae8a4acbd6d05a","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-7ce10652a7d44cf0","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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,         16;\n  2:       12,        136,         216;\n  3:       60,       1064,        3528,         3072;\n  4:      360,       8784,       45108,        81152,        48000;\n  5:     2520,      78768,      552924,      1530368,      1857600,        829440;\n  6:    20160,     771264,     6904584,     26004608,     48337600,      44084736,      15805440;\n  7:   181440,    8231616,    89906760,    430438528,   1085153600,    1510460928,    1102316544,     330301440;\n  8:  1814400,   95379840,  1232448048,   7181401600,  22957619600,   43231790592,   48029458176,   29218701312,   7524679680;\n  9: 19958400, 1194330240, 17849021328, 122815792640, 476952615600, 1136729140992, 1705189450496, 1576052785152, 822369853440, 185794560000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(8*n^3+16*n^2+8*n)*x-(36*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^3+16*n^2+8*n)*x^2+(16*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(8*n^2+8*n)*x)*P(n-1,x) - (8*n^2+8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x - (36*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^3 + 16*n^2 + 8*n)*x^2 + (16*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*P[n - 1, x] - (8*n^2 + 8*n)*x^2*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x - (36*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^3 + 16*n^2 + 8*n)*x^2 + (16*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*v[i - 1] - (8*n^2 + 8*n)*x^2*deriv(v[i - 1], x))/(n)); 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-2feb6dc01f424b0a","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-5f6e941a4a66ef73","e-abfc96fe738530bb"],"text":"A145324 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A145324 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}