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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":"58478fa0e0feed961299cdc944e6b802ef7153026ba31d3362c540f9df44002a","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-e7d0ffa461d9dbc4","role":"sequence","name":"Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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,  16;\n  2: 1,  32,   81;\n  3: 1,  48,  243,   256;\n  4: 1,  64,  486,  1024,   625;\n  5: 1,  80,  810,  2560,  3125,   1296;\n  6: 1,  96, 1215,  5120,  9375,   7776,   2401;\n  7: 1, 112, 1701,  8960, 21875,  27216,  16807,   4096;\n  8: 1, 128, 2268, 14336, 43750,  72576,  67228,  32768,  6561;\n  9: 1, 144, 2916, 21504, 78750, 163296, 201684, 147456, 59049, 10000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 1 + 16*x, P(2,x) = 1 + 32*x + 81*x^2, P(3,x) = 1 + 48*x + 243*x^2 + 256*x^3, P(4,x) = 1 + 64*x + 486*x^2 + 1024*x^3 + 625*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 16*x;\nP[2, x_] := 1 + 32*x + 81*x^2;\nP[3, x_] := 1 + 48*x + 243*x^2 + 256*x^3;\nP[4, x_] := 1 + 64*x + 486*x^2 + 1024*x^3 + 625*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 5*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; v[3] = 1 + 32*x + 81*x^2; v[4] = 1 + 48*x + 243*x^2 + 256*x^3; v[5] = 1 + 64*x + 486*x^2 + 1024*x^3 + 625*x^4; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 5*x)*v[i - 1]); 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-c8c19eba9d68881f","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-6f35c326bbf0f62d","e-e5d4e680eaaa219e"],"text":"A007318 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}