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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":"c8716c30b1f5de05caed9a254cba5254ab812b509c700cfd76b71e47be1f7348","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-f93ed6bfdfdae24a","role":"sequence","name":"theta_plus_one of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-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: 1,  4;\n  2: 1, 12,   9;\n  3: 1, 16,  36,   16;\n  4: 1, 24,  99,   96,   25;\n  5: 1, 28, 153,  272,  175,    36;\n  6: 1, 36, 270,  720,  750,   324,   49;\n  7: 1, 40, 351, 1200, 1875,  1404,  490,   64;\n  8: 1, 48, 522, 2304, 4875,  5184, 2842,  768,   81;\n  9: 1, 52, 630, 3232, 8475, 12204, 9898, 4480, 1053, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 12*x + 9*x^2, P(3,x) = 1 + 16*x + 36*x^2 + 16*x^3, P(4,x) = 1 + 24*x + 99*x^2 + 96*x^3 + 25*x^4, P(5,x) = 1 + 28*x + 153*x^2 + 272*x^3 + 175*x^4 + 36*x^5.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 12*x + 9*x^2;\nP[3, x_] := 1 + 16*x + 36*x^2 + 16*x^3;\nP[4, x_] := 1 + 24*x + 99*x^2 + 96*x^3 + 25*x^4;\nP[5, x_] := 1 + 28*x + 153*x^2 + 272*x^3 + 175*x^4 + 36*x^5;\nP[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 12*x + 9*x^2; v[4] = 1 + 16*x + 36*x^2 + 16*x^3; v[5] = 1 + 24*x + 99*x^2 + 96*x^3 + 25*x^4; v[6] = 1 + 28*x + 153*x^2 + 272*x^3 + 175*x^4 + 36*x^5; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 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-4e0b19699f4cf0db","steps":["\u03b8+1","\u03b8+1"],"edges":["e-9d3ba3f44f61794d","e-91f3668b1013eb5d"],"text":"A026374 \u2192 \u03b8+1 \u2192 \u03b8+1"},"headline":"A026374 \u2192 \u03b8+1 \u2192 \u03b8+1","chain":null}