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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":"617f160a7ae627d35bbdc8c1e72541916f8b9dc69b00a046c3ba22f96890646a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a4555bce014ca1b5","role":"sequence","name":"derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   0;\n   2:   1;\n   3:   5,     8;\n   4:  14,    56,    27;\n   5:  30,   216,   243,     64;\n   6:  55,   616,  1188,    704,    125;\n   7:  91,  1456,  4212,   4160,   1625,    216;\n   8: 140,  3024, 12150,  17600,  11250,   3240,   343;\n   9: 204,  5712, 30294,  59840,  55250,  25704,  5831,  512;\n  10: 285, 10032, 67716, 173888, 216125, 143640, 52136, 9728, 729;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-28*n+24+(4*n^4-26*n^3+58*n^2-50*n+12)*x+(2*n^6-19*n^5+74*n^4-151*n^3+170*n^2-100*n+24)*x^2)*P(n,x) = (8*n^2-4*n+(4*n^4-14*n^3+26*n^2-18*n)*x+(2*n^6-13*n^5+34*n^4-43*n^3+26*n^2-6*n)*x^2+(n^6-5*n^5+10*n^4-10*n^3+5*n^2-n)*x^3)*P(n-1,x) + ((16*n-16)*x+(8*n^3-24*n^2+28*n-12)*x^2+(4*n^5-20*n^4+42*n^3-46*n^2+26*n-6)*x^3+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (4*n^4 - 14*n^3 + 26*n^2 - 18*n)*x + (2*n^6 - 13*n^5 + 34*n^4 - 43*n^3 + 26*n^2 - 6*n)*x^2 + (n^6 - 5*n^5 + 10*n^4 - 10*n^3 + 5*n^2 - n)*x^3)*P[n - 1, x] + ((16*n - 16)*x + (8*n^3 - 24*n^2 + 28*n - 12)*x^2 + (4*n^5 - 20*n^4 + 42*n^3 - 46*n^2 + 26*n - 6)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)*D[P[n - 1, x], x])/(8*n^2 - 28*n + 24 + (4*n^4 - 26*n^3 + 58*n^2 - 50*n + 12)*x + (2*n^6 - 19*n^5 + 74*n^4 - 151*n^3 + 170*n^2 - 100*n + 24)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 4*n + (4*n^4 - 14*n^3 + 26*n^2 - 18*n)*x + (2*n^6 - 13*n^5 + 34*n^4 - 43*n^3 + 26*n^2 - 6*n)*x^2 + (n^6 - 5*n^5 + 10*n^4 - 10*n^3 + 5*n^2 - n)*x^3)*v[i - 1] + ((16*n - 16)*x + (8*n^3 - 24*n^2 + 28*n - 12)*x^2 + (4*n^5 - 20*n^4 + 42*n^3 - 46*n^2 + 26*n - 6)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)*deriv(v[i - 1], x))/(8*n^2 - 28*n + 24 + (4*n^4 - 26*n^3 + 58*n^2 - 50*n + 12)*x + (2*n^6 - 19*n^5 + 74*n^4 - 151*n^3 + 170*n^2 - 100*n + 24)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-05948cabe35ede38","e-ad13fcb28bdf8486"],"text":"A111125 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A111125 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}