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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":"a3c028bc7a5174c58b7590b33dc1d7d5b18dee8dd0257fb2d2582b60a8d37c82","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-67b93749ab954f15","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.","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":[],"oeis":[],"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,      12;\n   3: 1,      60,        135;\n   4: 1,     252,       1890,        1680;\n   5: 1,    1020,      19845,       50400,         23625;\n   6: 1,    4092,     190080,     1053360,       1299375,        374220;\n   7: 1,   16380,    1756755,    19219200,      47297250,      34054020,       6621615;\n   8: 1,   65532,   15995070,   329369040,    1452701250,    1975133160,     927026100,     129729600;\n   9: 1,  262140,  144692865,  5468954400,   40949283375,   94115581560,   80373162870,   26464838400,   2791213425;\n  10: 1, 1048572, 1305184860, 89303892720, 1100639255625, 4036797584820, 5603607909900, 3268407542400, 795495826125, 65472907500;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 39*x^2*P(n-2,x) + (13*x^2+56*x^3)*d/dx P(n-2,x) + (25*x^3+12*x^4)*d^2/dx^2 P(n-2,x) + 6*x^4*d^3/dx^3 P(n-2,x) + (1+12*x)*P(n-1,x) + (3*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[39*x^2*P[n - 2, x] + (13*x^2 + 56*x^3)*D[P[n - 2, x], x] + (25*x^3 + 12*x^4)*D[P[n - 2, x], {x, 2}] + 6*x^4*D[P[n - 2, x], {x, 3}] + (1 + 12*x)*P[n - 1, x] + (3*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = 39*x^2*v[i - 2] + (13*x^2 + 56*x^3)*deriv(v[i - 2], x) + (25*x^3 + 12*x^4)*deriv(deriv(v[i - 2], x), x) + 6*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 12*x)*v[i - 1] + (3*x - 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-67b93749ab954f15","steps":["(\u03b8+1)\u00b2"],"edges":["e-80a0b5998b7a3415"],"text":"A156289 \u2192 (\u03b8+1)\u00b2"},"headline":"A156289 \u2192 (\u03b8+1)\u00b2","chain":null}