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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":"6fc61503bffe6b06748ca21d801c1abb19afcb656e816fadd78700b1b8e6dc1e","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-0486a9621eaf19a3","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 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:  2,  12;\n  2:  3,  32,   45;\n  3:  4,  60,  162,   112;\n  4:  5,  96,  378,   512,   225;\n  5:  6, 140,  720,  1440,  1250,   396;\n  6:  7, 192, 1215,  3200,  4125,  2592,   637;\n  7:  8, 252, 1890,  6160, 10500,  9828,  4802,   960;\n  8:  9, 320, 2772, 10752, 22750, 28224, 20580,  8192,  1377;\n  9: 10, 396, 3888, 17472, 44100, 68040, 65856, 39168, 13122, 1900;\n","recurrence_sentence":"The row polynomials satisfy (n+(3*n^2+4*n-4)*x+(n^3+8*n^2-8*n+2)*x^2+(2*n^3-n^2)*x^3)*P(n,x) = (n+1+(3*n^2+11*n+4)*x+(n^3+14*n^2+21*n+6)*x^2+(3*n^3+16*n^2+15*n+4)*x^3+(2*n^3+5*n^2+4*n+1)*x^4)*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[((n + 1 + (3*n^2 + 11*n + 4)*x + (n^3 + 14*n^2 + 21*n + 6)*x^2 + (3*n^3 + 16*n^2 + 15*n + 4)*x^3 + (2*n^3 + 5*n^2 + 4*n + 1)*x^4)*P[n - 1, x])/(n + (3*n^2 + 4*n - 4)*x + (n^3 + 8*n^2 - 8*n + 2)*x^2 + (2*n^3 - n^2)*x^3)]];\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] = ((n + 1 + (3*n^2 + 11*n + 4)*x + (n^3 + 14*n^2 + 21*n + 6)*x^2 + (3*n^3 + 16*n^2 + 15*n + 4)*x^3 + (2*n^3 + 5*n^2 + 4*n + 1)*x^4)*v[i - 1])/(n + (3*n^2 + 4*n - 4)*x + (n^3 + 8*n^2 - 8*n + 2)*x^2 + (2*n^3 - n^2)*x^3)); 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":["\u03b8+1+n","(\u03b8+1)\u00b2"],"edges":["e-dd616c2789f09167","e-ff38e1ff2562ebb3"],"text":"A007318 \u2192 \u03b8+1+n \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 \u03b8+1+n \u2192 (\u03b8+1)\u00b2","chain":null}