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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":"d3e6577b84660ebcd141da0ce31772ba7e3d57b8415e7c1dab7a067c18eca0da","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-4e0b19699f4cf0db","role":"sequence","name":"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":[{"id":"l-1ad370b769f40467","kind":"oeis_match","node_a_id":"n-f93ed6bfdfdae24a","node_b_id":null,"fingerprint":null,"oeis_id":"A026950","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A026950","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A026950","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f93ed6bfdfdae24a","oeis_id":"A026950","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":8261}],"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,  2;\n  2: 1,  6,   3;\n  3: 1,  8,  12,   4;\n  4: 1, 12,  33,  24,    5;\n  5: 1, 14,  51,  68,   35,    6;\n  6: 1, 18,  90, 180,  150,   54,    7;\n  7: 1, 20, 117, 300,  375,  234,   70,   8;\n  8: 1, 24, 174, 576,  975,  864,  406,  96,   9;\n  9: 1, 26, 210, 808, 1695, 2034, 1414, 560, 117, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+6*x+11*x^2+6*x^3+x^4)*P(n-4,x) + (2+6*x+2*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 6*x + 3*x^2, P(3,x) = 1 + 8*x + 12*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 6*x + 3*x^2;\nP[3, x_] := 1 + 8*x + 12*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 11*x^2 + 6*x^3 + x^4)*P[n - 4, x] + (2 + 6*x + 2*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 + 2*x; v[3] = 1 + 6*x + 3*x^2; v[4] = 1 + 8*x + 12*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 6*x + 11*x^2 + 6*x^3 + x^4)*v[i - 4] + (2 + 6*x + 2*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"],"edges":["e-9d3ba3f44f61794d"],"text":"A026374 \u2192 \u03b8+1"},"headline":"A026374 \u2192 \u03b8+1","chain":null}