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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":"b60ad7d216eba29bfb4efed68fd642f739d61628051967b9d4640353820bc446","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-e5bdf85b655e989a","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+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:     2;\n  2:     7,     8;\n  3:    20,    32;\n  4:    61,   144,     48;\n  5:   182,   544,    288;\n  6:   547,  2008,   1584,   256;\n  7:  1640,  7104,   7296,  2048;\n  8:  4921, 24608,  31392, 13312,  1280;\n  9: 14762, 83648, 127296, 71680, 12800;\n","recurrence_sentence":"The row polynomials satisfy (12*n+(8*n^2+11*n)*x+(2*n^3+2*n^2)*x^2)*P(n,x) = (60*n-36+(52*n^2+91*n-105)*x+(10*n^3+68*n^2+18*n-88)*x^2+(8*n^3+16*n^2-8*n-16)*x^3)*P(n-1,x) - (36+(48*n+150)*x+(12*n^2+148*n+226)*x^2+(28*n^2+148*n+144)*x^3+(16*n^2+48*n+32)*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[((60*n - 36 + (52*n^2 + 91*n - 105)*x + (10*n^3 + 68*n^2 + 18*n - 88)*x^2 + (8*n^3 + 16*n^2 - 8*n - 16)*x^3)*P[n - 1, x] - (36 + (48*n + 150)*x + (12*n^2 + 148*n + 226)*x^2 + (28*n^2 + 148*n + 144)*x^3 + (16*n^2 + 48*n + 32)*x^4)*D[P[n - 1, x], x])/(12*n + (8*n^2 + 11*n)*x + (2*n^3 + 2*n^2)*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] = ((60*n - 36 + (52*n^2 + 91*n - 105)*x + (10*n^3 + 68*n^2 + 18*n - 88)*x^2 + (8*n^3 + 16*n^2 - 8*n - 16)*x^3)*v[i - 1] - (36 + (48*n + 150)*x + (12*n^2 + 148*n + 226)*x^2 + (28*n^2 + 148*n + 144)*x^3 + (16*n^2 + 48*n + 32)*x^4)*deriv(v[i - 1], x))/(12*n + (8*n^2 + 11*n)*x + (2*n^3 + 2*n^2)*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-cf38558d99f4fe37","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-7b35457a4f9e8cf8","e-49e52a0f4849fc32"],"text":"A229032 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A229032 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}