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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":"245b3a310fa76edff4fd7342ce48a4172426a3f350521c833ddcafc2fff271e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9516b04042fe64c0","role":"sequence","name":"Coefficient triangle of certain polynomials N(4; m,x).","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: 1,  20;\n  2: 1,  48,   135;\n  3: 1,  84,   567,    560;\n  4: 1, 128,  1512,   3584,    1750;\n  5: 1, 180,  3240,  13440,   15750,    4536;\n  6: 1, 240,  6075,  38400,   78750,   54432,   10290;\n  7: 1, 308, 10395,  92400,  288750,  349272,  158466,   21120;\n  8: 1, 384, 16632, 197120,  866250, 1596672, 1267728,  405504,  40095;\n  9: 1, 468, 25272, 384384, 2252250, 5837832, 7063056, 3953664, 938223, 71500;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n-(4*n^4+30*n^3+52*n^2-16*n)*x+(n^6+12*n^5+48*n^4+64*n^3)*x^2)*P(n,x) = (n^2+4*n-(4*n^4+30*n^3+40*n^2-56*n-48)*x+(n^6+12*n^5+27*n^4-106*n^3-441*n^2-552*n-216)*x^2+(3*n^6+42*n^5+219*n^4+564*n^3+840*n^2+640*n+192)*x^3)*P(n-1,x) + ((2*n+4)*x-(8*n^3+56*n^2+96*n+48)*x^2+(2*n^5+26*n^4+124*n^3+272*n^2+286*n+108)*x^3-(2*n^5+26*n^4+116*n^3+216*n^2+192*n+64)*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^2 + 4*n - (4*n^4 + 30*n^3 + 40*n^2 - 56*n - 48)*x + (n^6 + 12*n^5 + 27*n^4 - 106*n^3 - 441*n^2 - 552*n - 216)*x^2 + (3*n^6 + 42*n^5 + 219*n^4 + 564*n^3 + 840*n^2 + 640*n + 192)*x^3)*P[n - 1, x] + ((2*n + 4)*x - (8*n^3 + 56*n^2 + 96*n + 48)*x^2 + (2*n^5 + 26*n^4 + 124*n^3 + 272*n^2 + 286*n + 108)*x^3 - (2*n^5 + 26*n^4 + 116*n^3 + 216*n^2 + 192*n + 64)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n - (4*n^4 + 30*n^3 + 52*n^2 - 16*n)*x + (n^6 + 12*n^5 + 48*n^4 + 64*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^2 + 4*n - (4*n^4 + 30*n^3 + 40*n^2 - 56*n - 48)*x + (n^6 + 12*n^5 + 27*n^4 - 106*n^3 - 441*n^2 - 552*n - 216)*x^2 + (3*n^6 + 42*n^5 + 219*n^4 + 564*n^3 + 840*n^2 + 640*n + 192)*x^3)*v[i - 1] + ((2*n + 4)*x - (8*n^3 + 56*n^2 + 96*n + 48)*x^2 + (2*n^5 + 26*n^4 + 124*n^3 + 272*n^2 + 286*n + 108)*x^3 - (2*n^5 + 26*n^4 + 116*n^3 + 216*n^2 + 192*n + 64)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n - (4*n^4 + 30*n^3 + 52*n^2 - 16*n)*x + (n^6 + 12*n^5 + 48*n^4 + 64*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-9516b04042fe64c0","steps":["(\u03b8+1)\u00b2"],"edges":["e-4f5eaf3b19bd0a90"],"text":"A062264 \u2192 (\u03b8+1)\u00b2"},"headline":"A062264 \u2192 (\u03b8+1)\u00b2","chain":null}