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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":"8834fba98580e2f2c54700359700d25f7ddf4ac81db334ed878dadab791292f2","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-3bdf31527eaa2fb3","role":"sequence","name":"basis_map_A007318 of Pascal-(1,2,1) array 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:    1;\n  1:    2,      4;\n  2:    6,     24,      9;\n  3:   16,     96,     90,     16;\n  4:   44,    352,    522,    224,     25;\n  5:  120,   1200,   2448,   1728,    450,     36;\n  6:  328,   3936,  10224,  10112,   4350,    792,     49;\n  7:  896,  12544,  39600,  50560,  30800,   9216,   1274,    64;\n  8: 2448,  39168, 145584, 228096, 180900,  76896,  17346,  1920,   81;\n  9: 6688, 120384, 514944, 956928, 937800, 521424, 166992, 29952, 2754, 100;\n","recurrence_sentence":"The row polynomials satisfy (48+(144*n-24)*x+(132*n^2-12*n-20)*x^2+(36*n^3+30*n^2-30*n-6)*x^3+(3*n^4+6*n^3-7*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (96+(288*n+336)*x+(264*n^2+792*n+440)*x^2+(72*n^3+540*n^2+780*n+260)*x^3+(6*n^4+108*n^3+334*n^2+300*n+60)*x^4+(6*n^4+34*n^3+50*n^2+14*n-4)*x^5-(2*n^3+8*n^2+10*n+4)*x^6)*P(n-2,x) + (96+(288*n+144)*x+(264*n^2+384*n+32)*x^2+(72*n^3+300*n^2+168*n-80)*x^3+(6*n^4+60*n^3+100*n^2-4*n-42)*x^4+(3*n^4+10*n^3-12*n-5)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((96 + (288*n + 336)*x + (264*n^2 + 792*n + 440)*x^2 + (72*n^3 + 540*n^2 + 780*n + 260)*x^3 + (6*n^4 + 108*n^3 + 334*n^2 + 300*n + 60)*x^4 + (6*n^4 + 34*n^3 + 50*n^2 + 14*n - 4)*x^5 - (2*n^3 + 8*n^2 + 10*n + 4)*x^6)*P[n - 2, x] + (96 + (288*n + 144)*x + (264*n^2 + 384*n + 32)*x^2 + (72*n^3 + 300*n^2 + 168*n - 80)*x^3 + (6*n^4 + 60*n^3 + 100*n^2 - 4*n - 42)*x^4 + (3*n^4 + 10*n^3 - 12*n - 5)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(48 + (144*n - 24)*x + (132*n^2 - 12*n - 20)*x^2 + (36*n^3 + 30*n^2 - 30*n - 6)*x^3 + (3*n^4 + 6*n^3 - 7*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((96 + (288*n + 336)*x + (264*n^2 + 792*n + 440)*x^2 + (72*n^3 + 540*n^2 + 780*n + 260)*x^3 + (6*n^4 + 108*n^3 + 334*n^2 + 300*n + 60)*x^4 + (6*n^4 + 34*n^3 + 50*n^2 + 14*n - 4)*x^5 - (2*n^3 + 8*n^2 + 10*n + 4)*x^6)*v[i - 2] + (96 + (288*n + 144)*x + (264*n^2 + 384*n + 32)*x^2 + (72*n^3 + 300*n^2 + 168*n - 80)*x^3 + (6*n^4 + 60*n^3 + 100*n^2 - 4*n - 42)*x^4 + (3*n^4 + 10*n^3 - 12*n - 5)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(48 + (144*n - 24)*x + (132*n^2 - 12*n - 20)*x^2 + (36*n^3 + 30*n^2 - 30*n - 6)*x^3 + (3*n^4 + 6*n^3 - 7*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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-f25cd955a876cde3","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-6f8e74d475c3b6cc","e-622121e99cd940f5"],"text":"A081577 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A081577 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}