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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":"66bbd944cb0c911a339d4da3868eb9ff356023391aa7e499e0e28e136c70d2ff","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-3c983d83f25bade9","role":"sequence","name":"shift_x_plus_one 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:     1;\n  1:     4,       8;\n  2:    14,      56,      27;\n  3:    48,     288,     288,       64;\n  4:   165,    1320,    2025,      960,      125;\n  5:   572,    5720,   11880,     8800,     2500,     216;\n  6:  2002,   24024,   63063,    64064,    28875,    5544,     343;\n  7:  7072,   99008,  314496,   407680,   254800,   78624,   10976,    512;\n  8: 25194,  403104, 1503684,  2376192,  1911000,  825552,  187278,  19968,   729;\n  9: 90440, 1627920, 6976800, 13023360, 12852000, 7197120, 2304960, 403200, 34020, 1000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+32*n^3+80*n^2+64*n+(6*n^5+45*n^4+98*n^3+48*n^2-32*n)*x+(n^6+8*n^5+20*n^4+16*n^3)*x^2)*P(n,x) = (16*n^4+104*n^3+248*n^2+256*n+96+(24*n^5+192*n^4+574*n^3+778*n^2+452*n+80)*x+(4*n^6+56*n^5+254*n^4+500*n^3+452*n^2+194*n+40)*x^2+(3*n^6+25*n^5+77*n^4+113*n^3+92*n^2+42*n+8)*x^3)*P(n-1,x) - ((8*n^3+40*n^2+64*n+32)*x+(12*n^4+80*n^3+187*n^2+179*n+60)*x^2+(2*n^5+28*n^4+118*n^3+205*n^2+149*n+36)*x^3+(2*n^5+16*n^4+46*n^3+58*n^2+34*n+8)*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[((16*n^4 + 104*n^3 + 248*n^2 + 256*n + 96 + (24*n^5 + 192*n^4 + 574*n^3 + 778*n^2 + 452*n + 80)*x + (4*n^6 + 56*n^5 + 254*n^4 + 500*n^3 + 452*n^2 + 194*n + 40)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*P[n - 1, x] - ((8*n^3 + 40*n^2 + 64*n + 32)*x + (12*n^4 + 80*n^3 + 187*n^2 + 179*n + 60)*x^2 + (2*n^5 + 28*n^4 + 118*n^3 + 205*n^2 + 149*n + 36)*x^3 + (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 32*n^3 + 80*n^2 + 64*n + (6*n^5 + 45*n^4 + 98*n^3 + 48*n^2 - 32*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*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] = ((16*n^4 + 104*n^3 + 248*n^2 + 256*n + 96 + (24*n^5 + 192*n^4 + 574*n^3 + 778*n^2 + 452*n + 80)*x + (4*n^6 + 56*n^5 + 254*n^4 + 500*n^3 + 452*n^2 + 194*n + 40)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*v[i - 1] - ((8*n^3 + 40*n^2 + 64*n + 32)*x + (12*n^4 + 80*n^3 + 187*n^2 + 179*n + 60)*x^2 + (2*n^5 + 28*n^4 + 118*n^3 + 205*n^2 + 149*n + 36)*x^3 + (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 32*n^3 + 80*n^2 + 64*n + (6*n^5 + 45*n^4 + 98*n^3 + 48*n^2 - 32*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*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-0e6d847331141ca6","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-1d39f03bcbd1a8b9","e-35333eadfc17d700"],"text":"A145596 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A145596 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}