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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":"471feca79c50c6fda8b0312ea0276e9bfd36a1bb8fafb7dec9692540b487f60d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5c0c5bce3aee8449","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-f0037911e38db2e9","kind":"oeis_match","node_a_id":"n-012ae01d9db4fd27","node_b_id":null,"fingerprint":null,"oeis_id":"A262442","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A262442","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A262442","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   2;\n   3: 1,   8,    3;\n   4: 1,  18,   30,     4;\n   5: 1,  32,  126,    80,      5;\n   6: 1,  50,  360,   560,    175,      6;\n   7: 1,  72,  825,  2400,   1890,    336,      7;\n   8: 1,  98, 1638,  7700,  11550,   5292,    588,     8;\n   9: 1, 128, 2940, 20384,  50050,  44352,  12936,   960,    9;\n  10: 1, 162, 4896, 47040, 171990, 252252, 144144, 28512, 1485, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (4*n^3-10*n^2+8*n-2+(36*n^3-132*n^2+152*n-52)*x)*P(n-1,x) + ((8*n^2-18*n+8)*x-(52*n^2-134*n+62)*x^2)*d/dx P(n-1,x) - ((5*n-3)*x^2-(20*n-12)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 10*n^2 + 8*n - 2 + (36*n^3 - 132*n^2 + 152*n - 52)*x)*P[n - 1, x] + ((8*n^2 - 18*n + 8)*x - (52*n^2 - 134*n + 62)*x^2)*D[P[n - 1, x], x] - ((5*n - 3)*x^2 - (20*n - 12)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 10*n^2 + 8*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 10*n^2 + 8*n - 2 + (36*n^3 - 132*n^2 + 152*n - 52)*x)*v[i - 1] + ((8*n^2 - 18*n + 8)*x - (52*n^2 - 134*n + 62)*x^2)*deriv(v[i - 1], x) - ((5*n - 3)*x^2 - (20*n - 12)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 10*n^2 + 8*n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-79ebafb266da3b41","steps":["\u2299 C(n,k+1) (shift -1)","\u03b8+1"],"edges":["e-1491609a6fc5f8c9","e-fe1ef6d74823af06"],"text":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 \u03b8+1"},"headline":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 \u03b8+1","chain":null}