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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":"85cbf3a2fcad6fc6c8bcf2a82a27d80ceb9aa3c1d6844580fbaf29309f132703","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-aaf95b093ddbf844","role":"sequence","name":"gamma_polynomial of Pascal-(1,5,1) array.","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;\n  2: 1,  20;\n  3: 1,  40;\n  4: 1,  60,  225;\n  5: 1,  80,  675;\n  6: 1, 100, 1350,  2000;\n  7: 1, 120, 2250,  8000;\n  8: 1, 140, 3375, 20000, 15625;\n  9: 1, 160, 4725, 40000, 78125;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n+(15*n^3-10*n)*x+(25*n^4+25*n^3)*x^2)*P(n,x) = (n^2+n+(15*n^3+40*n^2-10*n-40)*x+(25*n^4+275*n^3+200*n^2-350*n-100)*x^2+(250*n^4+1000*n^3+750*n^2-1000*n-1000)*x^3)*P(n-1,x) - ((n+1)*x+(15*n^2+50*n+30)*x^2+(25*n^3+425*n^2+800*n+300)*x^3+(500*n^3+2500*n^2+4000*n+2000)*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 + n + (15*n^3 + 40*n^2 - 10*n - 40)*x + (25*n^4 + 275*n^3 + 200*n^2 - 350*n - 100)*x^2 + (250*n^4 + 1000*n^3 + 750*n^2 - 1000*n - 1000)*x^3)*P[n - 1, x] - ((n + 1)*x + (15*n^2 + 50*n + 30)*x^2 + (25*n^3 + 425*n^2 + 800*n + 300)*x^3 + (500*n^3 + 2500*n^2 + 4000*n + 2000)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (15*n^3 - 10*n)*x + (25*n^4 + 25*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 + n + (15*n^3 + 40*n^2 - 10*n - 40)*x + (25*n^4 + 275*n^3 + 200*n^2 - 350*n - 100)*x^2 + (250*n^4 + 1000*n^3 + 750*n^2 - 1000*n - 1000)*x^3)*v[i - 1] - ((n + 1)*x + (15*n^2 + 50*n + 30)*x^2 + (25*n^3 + 425*n^2 + 800*n + 300)*x^3 + (500*n^3 + 2500*n^2 + 4000*n + 2000)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (15*n^3 - 10*n)*x + (25*n^4 + 25*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-6ad01dc6e705cdb6","steps":["\u03b3","(\u03b8+1)\u00b2"],"edges":["e-90e54edd477f164c","e-d7a50a8a06e20ebd"],"text":"A081580 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2"},"headline":"A081580 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2","chain":null}