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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":"9f999c003e6491cc16e0c9d6e14eb87726afd444ee55930497b5d133659a7ef4","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-32779d2065847504","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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:  3,    8;\n  2:  6,   32,   30;\n  3: 10,   80,  150,    80;\n  4: 15,  160,  450,   480,   175;\n  5: 21,  280, 1050,  1680,  1225,    336;\n  6: 28,  448, 2100,  4480,  4900,   2688,   588;\n  7: 36,  672, 3780, 10080, 14700,  12096,  5292,   960;\n  8: 45,  960, 6300, 20160, 36750,  40320, 26460,  9600,  1485;\n  9: 55, 1320, 9900, 36960, 80850, 110880, 97020, 52800, 16335, 2200;\n","recurrence_sentence":"The row polynomials satisfy (3*n+(5*n^2+n)*x+(n^3+2*n^2)*x^2)*P(n,x) = (3*n+6+(5*n^2+19*n+18)*x+(n^3+11*n^2+27*n+18)*x^2+(n^3+6*n^2+11*n+6)*x^3)*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[((3*n + 6 + (5*n^2 + 19*n + 18)*x + (n^3 + 11*n^2 + 27*n + 18)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)*P[n - 1, x])/(3*n + (5*n^2 + n)*x + (n^3 + 2*n^2)*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] = ((3*n + 6 + (5*n^2 + 19*n + 18)*x + (n^3 + 11*n^2 + 27*n + 18)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)*v[i - 1])/(3*n + (5*n^2 + n)*x + (n^3 + 2*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-c8c19eba9d68881f","steps":["(1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2","\u03b8+1"],"edges":["e-6ba022abf95075ed","e-1ee9472cc71938b6"],"text":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 \u03b8+1"},"headline":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 \u03b8+1","chain":null}