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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":"5e5cec0751135d41e49b52f7e7bd6a1acba046513666ab8f6047717878ed6ba3","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-7bf0ce499426260c","role":"sequence","name":"derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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,    24;\n  3:  20,   160,    108;\n  4:  40,   600,   1080,     320;\n  5:  70,  1680,   5670,    4480,     750;\n  6: 112,  3920,  21168,   31360,   14000,    1512;\n  7: 168,  8064,  63504,  150528,  126000,   36288,    2744;\n  8: 240, 15120, 163296,  564480,  756000,  408240,   82320,   4608;\n  9: 330, 26400, 374220, 1774080, 3465000, 2993760, 1131900, 168960, 7290;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+12*n-16+(5*n^4+11*n^3-37*n^2+n+20)*x+(n^6+3*n^5-6*n^4-6*n^3+9*n^2+3*n-4)*x^2)*P(n,x) = (4*n^2+24*n+32+(5*n^4+26*n^3+59*n^2+2*n-72)*x+(n^6+6*n^5+30*n^4+28*n^3-51*n^2-42*n+48)*x^2+(3*n^6+12*n^5+9*n^4-12*n^3+16*n-8)*x^3)*P(n-1,x) + ((8*n+12)*x+(10*n^3+22*n^2-4*n-28)*x^2+(2*n^5+9*n^4+2*n^3-23*n^2-10*n+20)*x^3-(2*n^5+9*n^4+12*n^3-n^2-6*n+4)*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 + 24*n + 32 + (5*n^4 + 26*n^3 + 59*n^2 + 2*n - 72)*x + (n^6 + 6*n^5 + 30*n^4 + 28*n^3 - 51*n^2 - 42*n + 48)*x^2 + (3*n^6 + 12*n^5 + 9*n^4 - 12*n^3 + 16*n - 8)*x^3)*P[n - 1, x] + ((8*n + 12)*x + (10*n^3 + 22*n^2 - 4*n - 28)*x^2 + (2*n^5 + 9*n^4 + 2*n^3 - 23*n^2 - 10*n + 20)*x^3 - (2*n^5 + 9*n^4 + 12*n^3 - n^2 - 6*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 12*n - 16 + (5*n^4 + 11*n^3 - 37*n^2 + n + 20)*x + (n^6 + 3*n^5 - 6*n^4 - 6*n^3 + 9*n^2 + 3*n - 4)*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 + 24*n + 32 + (5*n^4 + 26*n^3 + 59*n^2 + 2*n - 72)*x + (n^6 + 6*n^5 + 30*n^4 + 28*n^3 - 51*n^2 - 42*n + 48)*x^2 + (3*n^6 + 12*n^5 + 9*n^4 - 12*n^3 + 16*n - 8)*x^3)*v[i - 1] + ((8*n + 12)*x + (10*n^3 + 22*n^2 - 4*n - 28)*x^2 + (2*n^5 + 9*n^4 + 2*n^3 - 23*n^2 - 10*n + 20)*x^3 - (2*n^5 + 9*n^4 + 12*n^3 - n^2 - 6*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 12*n - 16 + (5*n^4 + 11*n^3 - 37*n^2 + n + 20)*x + (n^6 + 3*n^5 - 6*n^4 - 6*n^3 + 9*n^2 + 3*n - 4)*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-0e6d847331141ca6","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-e77b91d6fbb51ebc","e-d436c1aaf9a14365"],"text":"A145596 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A145596 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}