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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":"752f885d9a4601670bd4f4ed400f204895d9dc17047f0f34b9432012052795d8","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-69a7d6f31c3b5900","role":"sequence","name":"derivative of Even row Pascal-square read by antidiagonals.","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:  2;\n  2:  4,   24;\n  3:  6,   80,   108;\n  4:  8,  168,   540,   320;\n  5: 10,  288,  1512,  2240,    750;\n  6: 12,  440,  3240,  8064,   7000,   1512;\n  7: 14,  624,  5940, 21120,  31500,  18144,   2744;\n  8: 16,  840,  9828, 45760,  99000,  99792,  41160,  4608;\n  9: 18, 1088, 15120, 87360, 250250, 370656, 271656, 84480, 7290;\n","recurrence_sentence":"The row polynomials satisfy (8*n^4-12*n^3+4*n^2+(48*n^5-168*n^4+204*n^3-98*n^2+12*n+2)*x+(32*n^6-144*n^5+264*n^4-252*n^3+132*n^2-36*n+4)*x^2)*P(n,x) = (8*n^4-4*n^3+(48*n^5-24*n^4-28*n^3-30*n^2-2*n)*x+(32*n^6+128*n^5-312*n^4+8*n^3+48*n^2+80*n+16)*x^2+(96*n^6-144*n^5-72*n^4+84*n^3+84*n^2-40*n-16)*x^3)*P(n-1,x) - (4*n^3*x+(24*n^4+4*n^3-2*n^2-7*n-1)*x^2+(16*n^5+96*n^4-60*n^3-16*n^2-22*n-2)*x^3+(64*n^5-48*n^3-32*n^2+24*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^4 - 4*n^3 + (48*n^5 - 24*n^4 - 28*n^3 - 30*n^2 - 2*n)*x + (32*n^6 + 128*n^5 - 312*n^4 + 8*n^3 + 48*n^2 + 80*n + 16)*x^2 + (96*n^6 - 144*n^5 - 72*n^4 + 84*n^3 + 84*n^2 - 40*n - 16)*x^3)*P[n - 1, x] - (4*n^3*x + (24*n^4 + 4*n^3 - 2*n^2 - 7*n - 1)*x^2 + (16*n^5 + 96*n^4 - 60*n^3 - 16*n^2 - 22*n - 2)*x^3 + (64*n^5 - 48*n^3 - 32*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(8*n^4 - 12*n^3 + 4*n^2 + (48*n^5 - 168*n^4 + 204*n^3 - 98*n^2 + 12*n + 2)*x + (32*n^6 - 144*n^5 + 264*n^4 - 252*n^3 + 132*n^2 - 36*n + 4)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^4 - 4*n^3 + (48*n^5 - 24*n^4 - 28*n^3 - 30*n^2 - 2*n)*x + (32*n^6 + 128*n^5 - 312*n^4 + 8*n^3 + 48*n^2 + 80*n + 16)*x^2 + (96*n^6 - 144*n^5 - 72*n^4 + 84*n^3 + 84*n^2 - 40*n - 16)*x^3)*v[i - 1] - (4*n^3*x + (24*n^4 + 4*n^3 - 2*n^2 - 7*n - 1)*x^2 + (16*n^5 + 96*n^4 - 60*n^3 - 16*n^2 - 22*n - 2)*x^3 + (64*n^5 - 48*n^3 - 32*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(8*n^4 - 12*n^3 + 4*n^2 + (48*n^5 - 168*n^4 + 204*n^3 - 98*n^2 + 12*n + 2)*x + (32*n^6 - 144*n^5 + 264*n^4 - 252*n^3 + 132*n^2 - 36*n + 4)*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-79ebafb266da3b41","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-032b739c94c6fe41","e-8e5d85bd56cda9bd"],"text":"A172431 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A172431 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}