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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":"9c7859b42ce7e23e13f2ff60e0e5e967c080975d8630509e2d6c57a0a0327866","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-f6268da8d9636534","role":"sequence","name":"Exponential Riordan array (1, x(1+x/2)).","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: 0, 4;\n  2: 0, 4,  9;\n  3: 0, 0, 27,  16;\n  4: 0, 0, 27,  96,   25;\n  5: 0, 0,  0, 240,  250,    36;\n  6: 0, 0,  0, 240, 1125,   540,    49;\n  7: 0, 0,  0,   0, 2625,  3780,  1029,    64;\n  8: 0, 0,  0,   0, 2625, 15120, 10290,  1792,   81;\n  9: 0, 0,  0,   0,    0, 34020, 61740, 24192, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^4+6*n^3+13*n^2+12*n+4-(4*n^3+18*n^2+14*n-4)*x+4*n^2*x^2)*P(n,x) = -(n^5+5*n^4+7*n^3-n^2-8*n-4-(5*n^4+28*n^3+49*n^2+42*n+20)*x+(8*n^3+34*n^2+58*n+28)*x^2-(4*n^2+8*n+4)*x^3)*P(n-1,x) + ((2*n^4+12*n^3+26*n^2+24*n+8)*x-(8*n^3+44*n^2+68*n+32)*x^2+(8*n^2+16*n+8)*x^3)*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^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (5*n^4 + 28*n^3 + 49*n^2 + 42*n + 20)*x + (8*n^3 + 34*n^2 + 58*n + 28)*x^2 - (4*n^2 + 8*n + 4)*x^3)*P[n - 1, x] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (8*n^3 + 44*n^2 + 68*n + 32)*x^2 + (8*n^2 + 16*n + 8)*x^3)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (4*n^3 + 18*n^2 + 14*n - 4)*x + 4*n^2*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^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (5*n^4 + 28*n^3 + 49*n^2 + 42*n + 20)*x + (8*n^3 + 34*n^2 + 58*n + 28)*x^2 - (4*n^2 + 8*n + 4)*x^3)*v[i - 1] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (8*n^3 + 44*n^2 + 68*n + 32)*x^2 + (8*n^2 + 16*n + 8)*x^3)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (4*n^3 + 18*n^2 + 14*n - 4)*x + 4*n^2*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-97b51734d39ede6c","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-01d50252f4aaa594","e-38acf07bf4ec9f74"],"text":"A100861 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A100861 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}