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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":"4fc49bba0949f6a041dd700f08747090f7aa061da706ad9ea72adca0df5e9bbe","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-aa9a15392d2524ca","role":"sequence","name":"derivative of 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:   0;\n  1:   2;\n  2:   8,    16;\n  3:  20,   120,     54;\n  4:  40,   480,    648,     128;\n  5:  70,  1400,   3780,    2240,     250;\n  6: 112,  3360,  15120,   17920,    6000,     432;\n  7: 168,  7056,  47628,   94080,   63000,   13608,    686;\n  8: 240, 13440, 127008,  376320,  420000,  181440,  27440,  1024;\n  9: 330, 23760, 299376, 1241856, 2079000, 1496880, 452760, 50688, 1458;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+8*n-12+(5*n^4+n^3-31*n^2+31*n-6)*x+(n^6-6*n^4+8*n^3-3*n^2)*x^2)*P(n,x) = (4*n^2+20*n+24+(5*n^4+16*n^3+35*n^2-28*n-52)*x+(n^6+3*n^5+21*n^4-15*n^3-42*n^2+32)*x^2+(3*n^6+3*n^5-6*n^4-9*n^3+9*n^2+4*n-4)*x^3)*P(n-1,x) + ((8*n+8)*x+(10*n^3+7*n^2-11*n-18)*x^2+(2*n^5+4*n^4-8*n^3-12*n^2+2*n+12)*x^3-(2*n^5+4*n^4+2*n^3-5*n^2-n+2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 20*n + 24 + (5*n^4 + 16*n^3 + 35*n^2 - 28*n - 52)*x + (n^6 + 3*n^5 + 21*n^4 - 15*n^3 - 42*n^2 + 32)*x^2 + (3*n^6 + 3*n^5 - 6*n^4 - 9*n^3 + 9*n^2 + 4*n - 4)*x^3)*P[n - 1, x] + ((8*n + 8)*x + (10*n^3 + 7*n^2 - 11*n - 18)*x^2 + (2*n^5 + 4*n^4 - 8*n^3 - 12*n^2 + 2*n + 12)*x^3 - (2*n^5 + 4*n^4 + 2*n^3 - 5*n^2 - n + 2)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 8*n - 12 + (5*n^4 + n^3 - 31*n^2 + 31*n - 6)*x + (n^6 - 6*n^4 + 8*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 20*n + 24 + (5*n^4 + 16*n^3 + 35*n^2 - 28*n - 52)*x + (n^6 + 3*n^5 + 21*n^4 - 15*n^3 - 42*n^2 + 32)*x^2 + (3*n^6 + 3*n^5 - 6*n^4 - 9*n^3 + 9*n^2 + 4*n - 4)*x^3)*v[i - 1] + ((8*n + 8)*x + (10*n^3 + 7*n^2 - 11*n - 18)*x^2 + (2*n^5 + 4*n^4 - 8*n^3 - 12*n^2 + 2*n + 12)*x^3 - (2*n^5 + 4*n^4 + 2*n^3 - 5*n^2 - n + 2)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 8*n - 12 + (5*n^4 + n^3 - 31*n^2 + 31*n - 6)*x + (n^6 - 6*n^4 + 8*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":["D","(\u03b8+1)\u00b2"],"edges":["e-0041de9aacba5ab3","e-9dfd8a093536c5d9"],"text":"A108838 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A108838 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}