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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":"59505bfc3bbb0b770f7cc6547cc08185256cdf2a1702d2fb61a0da32fe0a0951","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c963abfc0266d66c","role":"sequence","name":"Triangle of Dyck paths counted by number of long interior inclines.","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:  2;\n  1:  3,    8;\n  2:  4,   32,    18;\n  3:  5,   80,   135,     32;\n  4:  6,  160,   540,    384,     50;\n  5:  7,  280,  1575,   2240,    875,     72;\n  6:  8,  448,  3780,   8960,   7000,   1728,     98;\n  7:  9,  672,  7938,  28224,  36750,  18144,   3087,   128;\n  8: 10,  960, 15120,  75264, 147000, 120960,  41160,  5120,  162;\n  9: 11, 1320, 26730, 177408, 485100, 598752, 339570, 84480, 8019, 200;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n+3+(2*n^4+8*n^3+5*n^2-4*n-3)*x+(n^6+6*n^5+12*n^4+10*n^3+3*n^2)*x^2)*P(n,x) = (n^2+5*n+6+(2*n^4+10*n^3+23*n^2+15*n-6)*x+(n^6+7*n^5+26*n^4+41*n^3+16*n^2-11*n)*x^2+(3*n^6+19*n^5+44*n^4+43*n^3+16*n^2+3*n)*x^3)*P(n-1,x) + ((2*n+3)*x+(4*n^3+12*n^2+8*n-3)*x^2+(2*n^5+13*n^4+28*n^3+23*n^2+6*n+3)*x^3-(2*n^5+13*n^4+32*n^3+35*n^2+16*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (2*n^4 + 10*n^3 + 23*n^2 + 15*n - 6)*x + (n^6 + 7*n^5 + 26*n^4 + 41*n^3 + 16*n^2 - 11*n)*x^2 + (3*n^6 + 19*n^5 + 44*n^4 + 43*n^3 + 16*n^2 + 3*n)*x^3)*P[n - 1, x] + ((2*n + 3)*x + (4*n^3 + 12*n^2 + 8*n - 3)*x^2 + (2*n^5 + 13*n^4 + 28*n^3 + 23*n^2 + 6*n + 3)*x^3 - (2*n^5 + 13*n^4 + 32*n^3 + 35*n^2 + 16*n + 3)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + 3 + (2*n^4 + 8*n^3 + 5*n^2 - 4*n - 3)*x + (n^6 + 6*n^5 + 12*n^4 + 10*n^3 + 3*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (2*n^4 + 10*n^3 + 23*n^2 + 15*n - 6)*x + (n^6 + 7*n^5 + 26*n^4 + 41*n^3 + 16*n^2 - 11*n)*x^2 + (3*n^6 + 19*n^5 + 44*n^4 + 43*n^3 + 16*n^2 + 3*n)*x^3)*v[i - 1] + ((2*n + 3)*x + (4*n^3 + 12*n^2 + 8*n - 3)*x^2 + (2*n^5 + 13*n^4 + 28*n^3 + 23*n^2 + 6*n + 3)*x^3 - (2*n^5 + 13*n^4 + 32*n^3 + 35*n^2 + 16*n + 3)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + 3 + (2*n^4 + 8*n^3 + 5*n^2 - 4*n - 3)*x + (n^6 + 6*n^5 + 12*n^4 + 10*n^3 + 3*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-c963abfc0266d66c","steps":["(\u03b8+1)\u00b2"],"edges":["e-ade1c1d52bd3c716"],"text":"A108838 \u2192 (\u03b8+1)\u00b2"},"headline":"A108838 \u2192 (\u03b8+1)\u00b2","chain":null}