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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":"e04ddd01484566b294ef2c8dd97823c38239bde98edfbc10bfa6dbce00970495","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-f1bdd4180f3e6ae5","role":"sequence","name":"basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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,   8;\n  2: 1,  16,   36;\n  3: 2,  40,  144,   128;\n  4: 1,  32,  216,   512,   400;\n  5: 2,  72,  576,  1792,  2400,   1152;\n  6: 1,  48,  540,  2560,  6000,   6912,   3136;\n  7: 2, 104, 1296,  7040, 20000,  31104,  25088,   8192;\n  8: 1,  64, 1008,  7168, 28000,  64512,  87808,  65536,  20736;\n  9: 2, 136, 2304, 17920, 78400, 209664, 351232, 360448, 207360, 51200;\n","recurrence_sentence":"The row polynomials satisfy (8+(40*n-16)*x+(112*n^2-144*n+32)*x^2+(96*n^3-128*n^2)*x^3+(32*n^4-64*n^3+32*n^2)*x^4)*P(n,x) = -(24*n-40+(204*n^2-372*n+56)*x+(480*n^3-880*n^2+176*n+32)*x^2+(384*n^4-288*n^3-768*n^2+288*n+448)*x^3+(96*n^5+288*n^4-800*n^3-96*n^2+512*n+256)*x^4+(128*n^5-192*n^4-128*n^3+256*n^2+64*n-128)*x^5)*P(n-1,x) + (3+(36*n-8)*x+(132*n^2+4*n-44)*x^2+(144*n^3+320*n^2-240*n-96)*x^3+(48*n^4+448*n^3+32*n^2-304*n-144)*x^4+(160*n^4+320*n^3-224*n^2-192*n)*x^5+(128*n^4-128*n^2+64)*x^6)*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[(-(24*n - 40 + (204*n^2 - 372*n + 56)*x + (480*n^3 - 880*n^2 + 176*n + 32)*x^2 + (384*n^4 - 288*n^3 - 768*n^2 + 288*n + 448)*x^3 + (96*n^5 + 288*n^4 - 800*n^3 - 96*n^2 + 512*n + 256)*x^4 + (128*n^5 - 192*n^4 - 128*n^3 + 256*n^2 + 64*n - 128)*x^5)*P[n - 1, x] + (3 + (36*n - 8)*x + (132*n^2 + 4*n - 44)*x^2 + (144*n^3 + 320*n^2 - 240*n - 96)*x^3 + (48*n^4 + 448*n^3 + 32*n^2 - 304*n - 144)*x^4 + (160*n^4 + 320*n^3 - 224*n^2 - 192*n)*x^5 + (128*n^4 - 128*n^2 + 64)*x^6)*D[P[n - 1, x], x])/(8 + (40*n - 16)*x + (112*n^2 - 144*n + 32)*x^2 + (96*n^3 - 128*n^2)*x^3 + (32*n^4 - 64*n^3 + 32*n^2)*x^4)]];\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] = (-(24*n - 40 + (204*n^2 - 372*n + 56)*x + (480*n^3 - 880*n^2 + 176*n + 32)*x^2 + (384*n^4 - 288*n^3 - 768*n^2 + 288*n + 448)*x^3 + (96*n^5 + 288*n^4 - 800*n^3 - 96*n^2 + 512*n + 256)*x^4 + (128*n^5 - 192*n^4 - 128*n^3 + 256*n^2 + 64*n - 128)*x^5)*v[i - 1] + (3 + (36*n - 8)*x + (132*n^2 + 4*n - 44)*x^2 + (144*n^3 + 320*n^2 - 240*n - 96)*x^3 + (48*n^4 + 448*n^3 + 32*n^2 - 304*n - 144)*x^4 + (160*n^4 + 320*n^3 - 224*n^2 - 192*n)*x^5 + (128*n^4 - 128*n^2 + 64)*x^6)*deriv(v[i - 1], x))/(8 + (40*n - 16)*x + (112*n^2 - 144*n + 32)*x^2 + (96*n^3 - 128*n^2)*x^3 + (32*n^4 - 64*n^3 + 32*n^2)*x^4)); 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-ea2139b8dbf6bf8d","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-d9920fdddfeeaf81","e-5e5134c41a3dcd7e"],"text":"A133080 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A133080 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}