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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":"541d98b457634e37db840d36b0c5009e36b23145a9889924fa38d9116a045241","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-9358d848e03a2727","role":"sequence","name":"derivative of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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:  0;\n  1:  1;\n  2:  3,   8;\n  3:  5,  48,    27;\n  4:  7, 120,   270,    64;\n  5:  9, 224,   945,   960,    125;\n  6: 11, 360,  2268,  4480,   2625,    216;\n  7: 13, 528,  4455, 13440,  15750,   6048,    343;\n  8: 15, 728,  7722, 31680,  57750,  45360,  12348,   512;\n  9: 17, 960, 12285, 64064, 160875, 199584, 113190, 23040, 729;\n","recurrence_sentence":"The row polynomials satisfy (16*n^4-56*n^3+68*n^2-34*n+6+(96*n^5-576*n^4+1320*n^3-1432*n^2+724*n-132)*x+(64*n^6-480*n^5+1488*n^4-2440*n^3+2232*n^2-1080*n+216)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(96*n^5-288*n^4+280*n^3-168*n^2+68*n-12)*x+(64*n^6+64*n^5-1024*n^4+1744*n^3-1124*n^2+456*n-72)*x^2+(192*n^6-864*n^5+1296*n^4-744*n^3+240*n^2-176*n+32)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(48*n^4-88*n^3+56*n^2-28*n+6)*x^2+(32*n^5+112*n^4-424*n^3+396*n^2-188*n+36)*x^3+(128*n^5-320*n^4+224*n^3-80*n^2+80*n-16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (96*n^5 - 288*n^4 + 280*n^3 - 168*n^2 + 68*n - 12)*x + (64*n^6 + 64*n^5 - 1024*n^4 + 1744*n^3 - 1124*n^2 + 456*n - 72)*x^2 + (192*n^6 - 864*n^5 + 1296*n^4 - 744*n^3 + 240*n^2 - 176*n + 32)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (48*n^4 - 88*n^3 + 56*n^2 - 28*n + 6)*x^2 + (32*n^5 + 112*n^4 - 424*n^3 + 396*n^2 - 188*n + 36)*x^3 + (128*n^5 - 320*n^4 + 224*n^3 - 80*n^2 + 80*n - 16)*x^4)*D[P[n - 1, x], x])/(16*n^4 - 56*n^3 + 68*n^2 - 34*n + 6 + (96*n^5 - 576*n^4 + 1320*n^3 - 1432*n^2 + 724*n - 132)*x + (64*n^6 - 480*n^5 + 1488*n^4 - 2440*n^3 + 2232*n^2 - 1080*n + 216)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (96*n^5 - 288*n^4 + 280*n^3 - 168*n^2 + 68*n - 12)*x + (64*n^6 + 64*n^5 - 1024*n^4 + 1744*n^3 - 1124*n^2 + 456*n - 72)*x^2 + (192*n^6 - 864*n^5 + 1296*n^4 - 744*n^3 + 240*n^2 - 176*n + 32)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (48*n^4 - 88*n^3 + 56*n^2 - 28*n + 6)*x^2 + (32*n^5 + 112*n^4 - 424*n^3 + 396*n^2 - 188*n + 36)*x^3 + (128*n^5 - 320*n^4 + 224*n^3 - 80*n^2 + 80*n - 16)*x^4)*deriv(v[i - 1], x))/(16*n^4 - 56*n^3 + 68*n^2 - 34*n + 6 + (96*n^5 - 576*n^4 + 1320*n^3 - 1432*n^2 + 724*n - 132)*x + (64*n^6 - 480*n^5 + 1488*n^4 - 2440*n^3 + 2232*n^2 - 1080*n + 216)*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-6758fe7729845be8","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-23ce7e4146b3d83c","e-eda578c87e9f870a"],"text":"A054142 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A054142 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}