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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":"c63e3eacc5fb9735d87d2bfc21e22508d82e6a4695a2f01239835ec4d9c8c56a","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-cf9f646a76b9c283","role":"sequence","name":"A triangular array related to A077028 and distributing the values of A007582.","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: 1,    8;\n  2: 1,   20,    36;\n  3: 1,   40,   153,    128;\n  4: 1,   76,   459,    784,    400;\n  5: 1,  144,  1206,   3136,   3225,    1152;\n  6: 1,  276,  2970,  10400,  16125,   11556,    3136;\n  7: 1,  536,  7047,  31040,  64375,   69336,   37681,   8192;\n  8: 1, 1052, 16317,  86576, 224875,  323316,  263767, 114752,  20736;\n  9: 1, 2080, 37116, 230272, 718550, 1292256, 1406300, 918016, 331857, 51200;\n","recurrence_sentence":"The row polynomials satisfy (4+(10*n-4)*x+(14*n^2-18*n+4)*x^2+(6*n^3-8*n^2)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = (4-(6*n-44)*x-(26*n^2-118*n+44)*x^2-(27*n^3-111*n^2+30*n+56)*x^3-(9*n^4-32*n^3-9*n^2+32*n+16)*x^4-(n^5-3*n^4-4*n^3+8*n^2+2*n-4)*x^5)*P(n-1,x) + (4*x+(17*n-16)*x^2+(20*n^2-6*n-36)*x^3+(8*n^3+13*n^2-35*n-18)*x^4+(n^4+8*n^3-11*n^2-12*n)*x^5+(n^4-4*n^2+2)*x^6)*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[((4 - (6*n - 44)*x - (26*n^2 - 118*n + 44)*x^2 - (27*n^3 - 111*n^2 + 30*n + 56)*x^3 - (9*n^4 - 32*n^3 - 9*n^2 + 32*n + 16)*x^4 - (n^5 - 3*n^4 - 4*n^3 + 8*n^2 + 2*n - 4)*x^5)*P[n - 1, x] + (4*x + (17*n - 16)*x^2 + (20*n^2 - 6*n - 36)*x^3 + (8*n^3 + 13*n^2 - 35*n - 18)*x^4 + (n^4 + 8*n^3 - 11*n^2 - 12*n)*x^5 + (n^4 - 4*n^2 + 2)*x^6)*D[P[n - 1, x], x])/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)]];\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] = ((4 - (6*n - 44)*x - (26*n^2 - 118*n + 44)*x^2 - (27*n^3 - 111*n^2 + 30*n + 56)*x^3 - (9*n^4 - 32*n^3 - 9*n^2 + 32*n + 16)*x^4 - (n^5 - 3*n^4 - 4*n^3 + 8*n^2 + 2*n - 4)*x^5)*v[i - 1] + (4*x + (17*n - 16)*x^2 + (20*n^2 - 6*n - 36)*x^3 + (8*n^3 + 13*n^2 - 35*n - 18)*x^4 + (n^4 + 8*n^3 - 11*n^2 - 12*n)*x^5 + (n^4 - 4*n^2 + 2)*x^6)*deriv(v[i - 1], x))/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)); 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-cf9f646a76b9c283","steps":["(\u03b8+1)\u00b2"],"edges":["e-68ef9961432d64bf"],"text":"A110552 \u2192 (\u03b8+1)\u00b2"},"headline":"A110552 \u2192 (\u03b8+1)\u00b2","chain":null}