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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":"a79cf42d147424fb430cc82228b1fab103b85af40a925c08f5484812210dbe91","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-cf38558d99f4fe37","role":"sequence","name":"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:  3,   16;\n  3:  4,   64;\n  4:  5,  160,   144;\n  5:  6,  320,   864;\n  6:  7,  560,  3024,   1024;\n  7:  8,  896,  8064,   8192;\n  8:  9, 1344, 18144,  36864,  6400;\n  9: 10, 1920, 36288, 122880, 64000;\n","recurrence_sentence":"The row polynomials satisfy (n-(10*n^3+2*n^2-4*n)*x+(4*n^5+8*n^4+4*n^3)*x^2)*P(n,x) = (n+1-(10*n^3+12*n^2-18*n+12)*x+(4*n^5+12*n^4-60*n^3-76*n^2+104*n+48)*x^2+(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64)*x^3)*P(n-1,x) + (2*x-(20*n^2+44*n+24)*x^2+(8*n^4+48*n^3+184*n^2+272*n+96)*x^3-(32*n^4+192*n^3+416*n^2+384*n+128)*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[((n + 1 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (4*n^5 + 12*n^4 - 60*n^3 - 76*n^2 + 104*n + 48)*x^2 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^3)*P[n - 1, x] + (2*x - (20*n^2 + 44*n + 24)*x^2 + (8*n^4 + 48*n^3 + 184*n^2 + 272*n + 96)*x^3 - (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^4)*D[P[n - 1, x], x])/(n - (10*n^3 + 2*n^2 - 4*n)*x + (4*n^5 + 8*n^4 + 4*n^3)*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] = ((n + 1 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (4*n^5 + 12*n^4 - 60*n^3 - 76*n^2 + 104*n + 48)*x^2 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^3)*v[i - 1] + (2*x - (20*n^2 + 44*n + 24)*x^2 + (8*n^4 + 48*n^3 + 184*n^2 + 272*n + 96)*x^3 - (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^4)*deriv(v[i - 1], x))/(n - (10*n^3 + 2*n^2 - 4*n)*x + (4*n^5 + 8*n^4 + 4*n^3)*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-75ccd0453a9b6cec","steps":["x \u21a6 x\u22121/4","(\u03b8+1)\u00b2"],"edges":["e-44df003225948a93","e-8f28dc5012800fde"],"text":"A011973 \u2192 x \u21a6 x\u22121/4 \u2192 (\u03b8+1)\u00b2"},"headline":"A011973 \u2192 x \u21a6 x\u22121/4 \u2192 (\u03b8+1)\u00b2","chain":null}